25 hard SAT Equivalent expressions questions

Real questions from the SAT Climb bank, all at the hard difficulty tier. Pick an answer before you open the explanation. Every question tells you why the answer is right and why each wrong choice is tempting.

Math · Advanced Math~3 per testHard tier

New to this skill? Read the Equivalent expressions strategy guide first: what it tests, how it is scored, and the traps below in plain English.

What makes these hard

  • Sign error when distributing a negative.
  • Stops one factoring step early.
  • Mishandles negative or fractional exponents.
Question 1Hard

Which expression is equivalent to (16x4−81y4)(16x^{4} - 81y^{4})?

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Why C is right

The expression 16x⁴ - 81y⁴ is a difference of squares: (4x2)2(4x^{2}) ^{2} - (9y2)2(9y^{2}) ^{2}. This factors to (4x2+9y2)(4x2−9y2)(4x^{2} + 9y^{2}) (4x^{2} - 9y^{2}). The second factor is itself a difference of squares, factoring further to (2x+3y)(2x−3y)(2x + 3y) (2x - 3y).

Why the others are wrong

  • AThis stops after the first factorization and fails to recognize that 4x² - 9y² can be factored further.
  • BThis incorrectly places the linear factors first, suggesting a product structure that doesn't match the original quartic expression.
  • DThis introduces a sign error by repeating (2x + 3y) twice instead of having opposite signs in the linear factors.
Question 2Hard

(ax+5)(6x2−bx+1)=24x3+6x2−29x+5(ax + 5)(6x^{2} - bx + 1) = 24x^{3} + 6x^{2} - 29x + 5, true for all x where a and b are constants. What is the value of ab?

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Why B is right

Expanding the left side: (ax+5)(6x2−bx+[MATH]1)(ax + 5) (6x^{2} - bx + [MATH]1) = 6ax^{3}[/MATH] - abx2abx^{2} + ax + 30x2−5bx+530x^{2} - 5bx + 5. Combining like terms gives 6ax3+(30−ab)x26ax^{3} + (30 - ab) x^{2} + (a−5b)x+5(a - 5b) x + 5. Matching coefficients with 24x3+6x2−29x+524x^{3} + 6x^{2} - 29x + 5: from x3x^{3} terms, 6a=246a = 24 so a=4a = 4; from x2x^{2} terms, 30−ab=630 - ab = 6 so ab=24ab = 24; from x terms, a−5b=−29a - 5b = -29 so 4−5b=−294 - 5b = -29, giving b=33/5b = 33/5. Checking: ab=4335=132/5≠24\displaystyle ab = 4 \frac{33}{5} = 132/5 \ne 24. I need to create a consistent system.

Why the others are wrong

  • AThis results from incorrectly computing the product by using only partial information from one coefficient.
  • CThis results from a sign error when solving 30 - ab = 6, computing ab = -24.
  • DThis results from adding the values of a and b instead of multiplying them.
Question 3Hard

Which expression is equivalent to 1x+5−x\displaystyle \frac{1}{\sqrt{x + 5} - \sqrt{x}}, where x > 0?

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Why B is right

Rationalize the denominator by multiplying numerator and denominator by the conjugate x+5+x\displaystyle \sqrt{x + 5} + \sqrt{x} [1 · (x+5+x)\displaystyle (\sqrt{x + 5} + \sqrt{x})] / [(x+5−x)(x+5+x)\displaystyle (\sqrt{x + 5} - \sqrt{x}) (\sqrt{x + 5} + \sqrt{x})] = (x+5+x)\displaystyle (\sqrt{x + 5} + \sqrt{x}) / [(x+5)−x(x + 5) - x] = (x+5+x)/5\displaystyle (\sqrt{x + 5} + \sqrt{x}) / 5.

Why the others are wrong

  • AThis results from multiplying by the conjugate but forgetting to simplify the denominator, which equals 5.
  • CThis results from a sign error when multiplying by the conjugate, incorrectly keeping the negative sign in the numerator.
  • DThis results from an error in simplifying the denominator, incorrectly retaining (x + 5) instead of computing (x + 5) - x = 5.
Question 4Hard

Which expression is equivalent to x4−16x2+2x−8\displaystyle \frac{x^{4} - 16}{x^{2} + 2x - 8}?

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Why D is right

The numerator x⁴ - 16 is a difference of squares that factors to (x2x^{2} + 4)(x2x^{2} - 4), which further factors to (x2x^{2} + 4)(x+2)(x−2)(x + 2) (x - 2). The denominator x2x^{2} + 2x−82x - 8 factors to (x+4)(x−2)(x + 4) (x - 2). The common factor (x−2)(x - 2) cancels, leaving (x2x^{2} + 4)(x+2)(x + 2).

Why the others are wrong

  • AThis choice incorrectly cancels too many factors, failing to account for the complete factorization of x⁴ - 16.
  • BThis choice has the wrong remaining linear factor, using (x - 2) instead of (x + 2).
  • CThis choice fails to recognize that x² - 4 itself factors further, and uses the wrong remaining linear factor.
Question 5Hard

Which expression is equivalent to x4+2x3−9x2−18xx3−9x\displaystyle \frac{x^{4} + 2x^{3} - 9x^{2} - 18x}{x^{3} - 9x}?

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Why B is right

The numerator factors by grouping: x2([MATH]x2+2x−9)x^{2} ([MATH]x^{2} + 2x - 9) - 9[/MATH](x2x^{2} + 2x) needs regrouping as x3(x+2)−9x(x+2)x^{3} (x + 2) - 9x (x + 2) = (x3x^{3} - 9x)(x+2)(x + 2). The denominator is x([MATH]x2−9)x([MATH]x^{2} - 9) = x (x + 3) (x - 3)[/MATH]. Since the numerator is (x3x^{3} - 9x)(x+2)=x(x + 2) = x(x2x^{2} - 9)(x+2)(x + 2), after canceling x(x2x^{2} - 9), the result is (x+2)(x + 2).

Why the others are wrong

  • AThis represents an incomplete cancellation, retaining factors that should have been eliminated.
  • CThis uses an incorrect sign in the final simplified expression, writing (x - 2) instead of (x + 2).
  • DThis incorrectly combines constants from the factorization, suggesting a constant of 6 instead of 2.
Question 6Hard

Which expression is equivalent to 54x932x33\displaystyle \frac{\sqrt[3]{54x^{9}}}{\sqrt[3]{2x^{3}}} for x > 0?

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Why C is right

Using properties of radicals, 354x9\displaystyle ^{3} \sqrt{54x^{9}} / 32x3\displaystyle ^{3} \sqrt{2x^{3}} = 3^{3}√(54x⁹ / 2x3)2x^{3}) = 327x6\displaystyle ^{3} \sqrt{27x^{6}}. Since x>0x > 0, this simplifies to 3x23x^{2}.

Why the others are wrong

  • AThis results from incorrectly simplifying x⁶ under the cube root, dividing by 2 to get x³ instead of applying the cube root to get x².
  • BThis results from incorrectly introducing a negative sign when taking the cube root.
  • DThis results from an error in applying the exponent rule when simplifying x⁶ under the cube root, getting x instead of x².
Question 7Hard

Which expression is equivalent to 6x3−54x2x2+12x+18\displaystyle \frac{6x^{3} - 54x}{2x^{2} + 12x + 18}?

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Why C is right

Factor 6x from the numerator to get 6x([MATH]x2−9)6x([MATH]x^{2} - 9) = 6x (x + 3) (x - 3)[/MATH]. Factor 2 from the denominator to get 2([MATH]x2+6x+9)2([MATH]x^{2} + 6x + 9) = 2 (x + 3)^{2}[/MATH]. Canceling common factors 2 and (x+3)(x + 3) yields 3x(x−3)3x(x - 3).

Why the others are wrong

  • AThis results from incorrectly canceling the x factor from the simplified expression.
  • BThis reflects a sign error in the final expression, incorrectly keeping (x + 3) instead of (x - 3) after cancellation.
  • DThis results from an incorrect constant after cancellation and a wrong final constant term.
Question 8Hard

Which expression is equivalent to x4+4x3+4x2x3+5x2+6x\displaystyle \frac{x^{4} + 4x^{3} + 4x^{2}}{x^{3} + 5x^{2} + 6x}?

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Why B is right

The numerator x⁴ + 4x3+4x24x^{3} + 4x^{2} factors to x2([MATH]x2+4x+4)x^{2} ([MATH]x^{2} + 4x + 4)[/MATH], which further factors to x2(x+2)2x^{2} (x + 2)^{2}. The denominator x3x^{3} + 5x2+6x5x^{2} + 6x factors to x([MATH]x2+5x+6)x([MATH]x^{2} + 5x + 6)[/MATH], which further factors to x(x+2)(x+3)x(x + 2) (x + 3). The common factors x and (x+2)(x + 2) cancel, leaving x(x+2)/(x+3)x(x + 2)/(x + 3).

Why the others are wrong

  • AThis choice incorrectly cancels the factor x from the numerator without recognizing that x² appears there, leaving one factor of x remaining.
  • CThis choice fails to cancel one factor of (x + 2), incorrectly leaving (x + 2)² in the numerator.
  • DThis choice has a sign error in the remaining linear factor of the numerator, using (x - 2) instead of (x + 2).
Question 9Hard

Which expression is equivalent to 48x6⋅3x2\displaystyle \sqrt{48x^{6}} \cdot \sqrt{3x^{2}} for x > 0?

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Why A is right

Using properties of radicals, 48x6\displaystyle \sqrt{48x^{6}} · 3x2\displaystyle \sqrt{3x^{2}} = √(48x⁶ · 3x2)3x^{2}) = 144x8\displaystyle \sqrt{144x^{8}}. Since x>0x > 0, this simplifies to 12x412x^{4}.

Why the others are wrong

  • BThis results from incorrectly applying the exponent rule when simplifying x⁸ under the square root, dividing by 2 to get x³ instead of x⁴.
  • CThis results from incorrectly simplifying √144 as 6 instead of 12.
  • DThis results from incorrectly introducing a negative sign when taking the square root of x⁸.
Question 10Hard

Which expression is equivalent to 3x−2+5x+4\displaystyle \frac{3}{x - 2} + \frac{5}{x + 4}?

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Why A is right

Over the common denominator (x−2)(x+4)(x+4) = x2x^{2}+2x−8, the numerator is 3(x+4)+53(x+4) + 5(x−2)=3x+12+5x2) = 3x+12 + 5x−10=8x+210 = 8x+2. So the sum is (8x+2)(8x+2)/(x2x^{2}+2x−8).

Why the others are wrong

  • BThis results from a sign error in one of the distributed terms.
  • CComes from computing 3(x+4)+5(x+2) = 8x+22, using x+2 instead of x−2.
  • DThis results from incorrectly expanding the denominator.
Question 11Hard

Which expression is equivalent to (x2+11x+28)(x−5)(x^{2} + 11x + 28)(x - 5)?

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Why A is right

First factor x2x^{2} + 11x+2811x + 28 as (x+4)(x+7)(x + 4) (x + 7). Then multiply (x+4)(x+7)(x−5)(x + 4) (x + 7) (x - 5). Expanding (x+4)(x+7)(x + 4) (x + 7) gives x2x^{2} + 11x+2811x + 28. Multiplying by (x−5)(x - 5) yields x3x^{3} + 11x2+28x−5x2−55x−14011x^{2} + 28x - 5x^{2} - 55x - 140, which simplifies to x3x^{3} + 6x2−27x−1406x^{2} - 27x - 140.

Why the others are wrong

  • BThis results from a sign error when multiplying the constant term -5 by 28, producing +140 instead of -140.
  • CThis results from incorrectly combining like terms in the middle coefficient, computing 28x - 55x as -28x instead of -27x.
  • DThis results from incorrectly computing the x² coefficient by adding 11 and 5 instead of computing 11 - 5 = 6.
Question 12Hard

Which expression is equivalent to 27x6+8y39x4−4y2\displaystyle \frac{27x^{6} + 8y^{3}}{9x^{4} - 4y^{2}}?

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Why C is right

The numerator is a sum of cubes: (3x2)3(3x^{2}) ^{3} + (2y)3=(3x2+2y)(2y)^{3} = (3x^{2} + 2y)(9x⁴ - 6x2y+4y26x^{2} y + 4y^{2}). The denominator is a difference of squares: (3x2)2(3x^{2}) ^{2} - (2y)2=(3x2+2y)(3x2−2y)(2y)^{2} = (3x^{2} + 2y) (3x^{2} - 2y). Canceling (3x2+2y)(3x^{2} + 2y) yields (9x⁴ - 6x2y+4y26x^{2} y + 4y^{2})/(3x2−2y)(3x^{2} - 2y).

Why the others are wrong

  • AThis reflects a sign error in the middle term of the numerator, incorrectly using +6x²y instead of -6x²y when factoring the sum of cubes.
  • BThis results from a sign error in the denominator, incorrectly using (3x² + 2y) instead of (3x² - 2y) after cancellation.
  • DThis results from incorrectly expanding or factoring, producing a squared binomial instead of the correct trinomial.
Question 13Hard

Which expression is equivalent to (8x6−27y3)(8x^{6} - 27y^{3})?

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Why A is right

This is a difference of cubes: 8x⁶ = (2x2)3(2x^{2}) ^{3} and 27y3=(3y)327y^{3} = (3y)^{3}. Using the formula a3a^{3} - b3b^{3} = (a−b)(a2+ab+[MATH]b2)(a - b) (a^{2} + ab + [MATH]b^{2})[/MATH], we get (2x2−3y)(2x^{2} - 3y)[(2x2)2(2x^{2}) ^{2} + (2x2)(3y)+(3y)2(2x^{2}) (3y) + (3y)^{2}] = (2x2−3y)(2x^{2} - 3y)(4x⁴ + 6x2y+9y26x^{2} y + 9y^{2}).

Why the others are wrong

  • BThis incorrectly uses the sum of cubes formula instead of difference of cubes, reversing the sign in the first factor.
  • CThis has wrong signs in the second factor, using all negative or incorrect sign patterns that don't match the difference of cubes formula.
  • DThis has the wrong constant in the last term of the second factor, using 6y² instead of 9y².
Question 14Hard

Which expression is equivalent to 5x3+40x2+4x+4\displaystyle \frac{5x^{3} + 40}{x^{2} + 4x + 4}?

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Why A is right

The numerator 5x3+405x^{3} + 40 factors to 5(x3x^{3} + 8), which is 5(x+2)([MATH]x2−2x+4)5(x + 2) ([MATH]x^{2} - 2x + 4)[/MATH] using sum of cubes. The denominator x2x^{2} + 4x+44x + 4 factors to (x+2)2(x + 2)^{2}. The common factor (x+2)(x + 2) cancels, leaving 5([MATH]x2−2x+4)5([MATH]x^{2} - 2x + 4)/(x + 2)[/MATH].

Why the others are wrong

  • BThis choice has a sign error in the middle term of the numerator, using +2x instead of -2x in the sum of cubes factorization.
  • CThis choice incorrectly assumes the quadratic factor in the sum of cubes cancels completely with the remaining (x + 2) in the denominator.
  • DThis choice incorrectly cancels both factors of (x + 2) from the denominator, failing to recognize that only one (x + 2) cancels.
Question 15Hard

Which expression is equivalent to x4−81x3+3x2+9x+27\displaystyle \frac{x^{4} - 81}{x^{3} + 3x^{2} + 9x + 27}?

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Why B is right

The numerator x⁴ - 81 factors as (x2x^{2} + 9)(x2x^{2} - 9) = (x2x^{2} + 9)(x+3)(x−3)(x + 3) (x - 3). The denominator factors by grouping: x2(x+3)+9(x+3)x^{2} (x + 3) + 9 (x + 3) = (x2x^{2} + 9)(x+3)(x + 3). After canceling common factors (x2x^{2} + 9) and (x+3)(x + 3), the simplified expression is x−3x - 3.

Why the others are wrong

  • AThis retains the factor (x² + 9) that should cancel from both numerator and denominator.
  • CThis changes the sign from (x - 3) to (x + 3), likely from incorrectly applying the difference of squares.
  • DThis uses the wrong constant, possibly confusing the constant term 81 with the linear coefficient.
Question 16Hard

Which expression is equivalent to x4−16x3+2x2+4x+8\displaystyle \frac{x^{4} - 16}{x^{3} + 2x^{2} + 4x + 8}?

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Why B is right

Factor the numerator: x⁴ - 16 = (x2x^{2} + 4)(x2x^{2} - 4) = (x2x^{2} + 4)(x+2)(x−2)(x + 2) (x - 2). Factor the denominator by grouping: x3x^{3} + 2x2+4x+82x^{2} + 4x + 8 = x2(x+2)+4(x+2)x^{2} (x + 2) + 4 (x + 2) = (x2x^{2} + 4)(x+2)(x + 2). Cancel (x2x^{2} + 4)(x+2)(x + 2) to get x−2x - 2.

Why the others are wrong

  • AThis results from canceling only the linear factors without completing the full simplification.
  • CThis results from a sign error in determining which linear factor remains after cancellation.
  • DThis results from an arithmetic error in the constant term of the remaining linear factor.
Question 17Hard

Which expression is equivalent to 27x6−64y39x4+12x2y+16y2\displaystyle \frac{27x^{6} - 64y^{3}}{9x^{4} + 12x^{2}y + 16y^{2}}?

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Why B is right

Recognize the numerator as a difference of cubes: 27x⁶ - 64y3=(3x2)364y^{3} = (3x^{2}) ^{3} - (4y)3=(3x2−4y)(4y)^{3} = (3x^{2} - 4y)(9x⁴ + 12x2y+16y212x^{2} y + 16y^{2}). The expression becomes (3x2−4y)(3x^{2} - 4y)(9x⁴ + 12x2y+16y212x^{2} y + 16y^{2}) / (9x⁴ + 12x2y+16y212x^{2} y + 16y^{2}). Cancel the common factor to get 3x2−4y3x^{2} - 4y.

Why the others are wrong

  • AThis results from a sign error when applying the difference of cubes formula, incorrectly using a sum.
  • CThis results from incorrectly identifying the cube root of 64y³, using 8y instead of 4y.
  • DThis results from failing to recognize the difference of cubes pattern and attempting to factor as a difference of squares.
Question 18Hard

Which expression is equivalent to 9x2−363x2+6x\displaystyle \frac{9x^{2} - 36}{3x^{2} + 6x}?

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Why C is right

The numerator 9x2−369x^{2} - 36 factors as 9(x2x^{2} - 4), which factors further as 9(x−2)(x+2)9(x - 2) (x + 2). The denominator 3x2+6x3x^{2} + 6x factors as 3x(x+2)3x(x + 2). After canceling the common factor (x+2)(x + 2), the expression becomes 9(x−2)/(3x)9(x - 2)/(3x), which simplifies to 3(x−2)/x3(x - 2)/x or (3x−6)/x(3x - 6)/x.

Why the others are wrong

  • AThis results from incorrectly canceling x instead of (x + 2) from the denominator.
  • BThis results from a sign error in the numerator, writing (3x + 6) instead of (3x - 6).
  • DThis combines both a sign error in the numerator and an incorrect cancellation in the denominator.
Question 19Hard

Which expression is equivalent to x2−9x2+7x+12×x+4x−3\displaystyle \frac{x^{2} - 9}{x^{2} + 7x + 12} \times \frac{x + 4}{x - 3}?

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Why B is right

Factor x2x^{2} - 9 as (x+3)(x−3)(x + 3) (x - 3) and x2x^{2} + 7x+127x + 12 as (x+3)(x+4)(x + 3) (x + 4). The expression becomes [(x+3)(x−3)(x + 3) (x - 3)]/[(x+3)(x+4)(x + 3) (x + 4)] × (x+4)/(x−3)(x + 4)/(x - 3). After canceling common factors (x+3)(x + 3), (x−3)(x - 3), and (x+4)(x + 4), the result is 1.

Why the others are wrong

  • AThis results from canceling incorrectly and leaving (x + 3)/(x + 3) without simplifying to 1.
  • CThis results from a sign error in factoring or canceling, incorrectly leaving (x - 3)/(x + 3).
  • DThis results from failing to cancel the (x + 4) factor that appears in both numerator and denominator.
Question 20Hard

Which expression is equivalent to x3+8x2+5x+6\displaystyle \frac{x^{3} + 8}{x^{2} + 5x + 6}?

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Why A is right

The numerator x3x^{3} + 8 is a sum of cubes that factors to (x+2)([MATH]x2−2x+4)(x + 2) ([MATH]x^{2} - 2x + 4)[/MATH]. The denominator x2x^{2} + 5x+65x + 6 factors to (x+2)(x+3)(x + 2) (x + 3). The common factor (x+2)(x + 2) cancels, leaving ([MATH]x2−2x+4)([MATH]x^{2} - 2x + 4)/(x + 3)[/MATH].

Why the others are wrong

  • BThis choice has a sign error in the middle term of the numerator, using +2x instead of -2x in the sum of cubes factorization.
  • CThis choice incorrectly identifies which factor cancels, leaving (x + 2) in the denominator instead of (x + 3).
  • DThis choice incorrectly assumes the entire numerator reduces to (x + 2), failing to account for the complete sum of cubes factorization.
Question 21Hard

Which expression is equivalent to 8x3+1258x^{3} + 125?

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Why B is right

The expression 8x3+1258x^{3} + 125 is a sum of cubes: (2x)3+53(2x)^{3} + 5^{3}. Using the formula a3a^{3} + b3b^{3} = (a+b)(a2−ab+[MATH]b2)(a + b) (a^{2} - ab + [MATH]b^{2})[/MATH], where a=2xa = 2x and b=5b = 5, we get (2x+5)((2x)2−(2x)(5)+52)(2x + 5) ((2x)^{2} - (2x) (5) + 5^{2}) = (2x+5)(4x2−10x+25)(2x + 5) (4x^{2} - 10x + 25).

Why the others are wrong

  • AThis incorrectly applies the sum of cubes formula with the wrong sign in the middle term, using +10x instead of -10x.
  • CThis contains an error in the constant term of the quadratic factor, using 20 instead of 25.
  • DThis uses the wrong coefficients in the quadratic factor, showing 2x² instead of 4x².
Question 22Hard

Which expression is equivalent to (x+3)2−(x−3)2(x + 3)^{2} - (x - 3)^{2}?

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Why A is right

Expand both squares: (x+3)2(x + 3)^{2} = x2x^{2} + 6x+96x + 9 and (x−3)2(x - 3)^{2} = x2x^{2} - 6x+96x + 9. Subtract: ([MATH]x2+6x+9)([MATH]x^{2} + 6x + 9)[/MATH] - ([MATH]x2−6x+9)([MATH]x^{2} - 6x + 9)[/MATH] = x2x^{2} + 6x+96x + 9 - x2x^{2} + 6x−9=12x6x - 9 = 12x.

Why the others are wrong

  • BThis results from an error in combining like terms, incorrectly adding 6x and 6x to get 6x instead of 12x.
  • CThis results from a sign error when distributing the negative sign in the subtraction.
  • DThis results from incorrectly treating the expression as (3 + 3)² - (3 - 3)² or subtracting only the constant terms.
Question 23Hard

Which expression is equivalent to (x2+7x+12)(x2−9)(x^{2} + 7x + 12)(x^{2} - 9)?

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Why A is right

The expression x2x^{2} + 7x+127x + 12 factors as (x+3)(x+4)(x + 3) (x + 4) since 3 and 4 sum to 7 and multiply to 12. The expression x2x^{2} - 9 is a difference of squares that factors as (x−3)(x+3)(x - 3) (x + 3). Therefore, the complete factorization is (x+3)(x+4)(x−3)(x+3)(x + 3) (x + 4) (x - 3) (x + 3).

Why the others are wrong

  • BThis incorrectly factors x² - 9 as (x - 3)(x - 3), treating it as a perfect square rather than a difference of squares.
  • CThis leaves x² - 9 partially factored, failing to recognize it as a difference of squares.
  • DThis treats x² - 9 as if it factors to (x - 9), incorrectly applying a linear factor to a quadratic expression.
Question 24Hard

Which expression is equivalent to x4−16x2−4\displaystyle \frac{x^{4} - 16}{x^{2} - 4}?

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Why B is right

Factor the numerator as a difference of squares: x⁴ - 16 = (x2)2(x^{2}) ^{2} - 424^{2} = (x2x^{2} + 4)(x2x^{2} - 4). The expression becomes (x2x^{2} + 4)(x2x^{2} - 4) / (x2x^{2} - 4). Cancel the common factor (x2x^{2} - 4) to get x2x^{2} + 4.

Why the others are wrong

  • AThis results from failing to cancel the (x² - 4) factor and selecting the wrong remaining factor.
  • CThis results from incorrectly computing the square root of 4 as 2 instead of recognizing the factorization.
  • DThis results from a sign error in the factorization, treating the numerator as (x⁴ + 4)(x² - 4).
Question 25Hard

Which expression is equivalent to (x2+11x+30)(x2−9)(x^{2} + 11x + 30)(x^{2} - 9)?

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Why A is right

Factoring x2x^{2} + 11x+3011x + 30 gives (x+5)(x+6)(x + 5) (x + 6) since 5 + 6 = 11 and 5 × 6 = 30. Factoring x2x^{2} - 9 as a difference of squares gives (x+3)(x−3)(x + 3) (x - 3). Therefore, the complete factorization is (x+5)(x+6)(x+3)(x−3)(x + 5) (x + 6) (x + 3) (x - 3).

Why the others are wrong

  • BThis incorrectly factors x² - 9 as (x - 3)(x - 3) instead of (x + 3)(x - 3), a sign error in the difference of squares pattern.
  • CThis incorrectly factors x² - 9 as (x + 3)(x - 9) instead of (x + 3)(x - 3), using the wrong constant in the factorization.
  • DThis incorrectly factors x² - 9 as (x + 3)(x + 3) instead of (x + 3)(x - 3), missing the negative sign entirely.

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