which product will result in a sum or difference of cubes?\n(x + 7)(x² - 7x + 14)\n(x + 8)(x² + 8x + 64)\n(x…

Answer:

C. $(x - 9)(x^{2}+9x + 81)$

Explanation:

Step1: Recall sum/difference - of - cubes formula

The sum of cubes formula is $(a + b)(a^{2}-ab + b^{2})=a^{3}+b^{3}$, and the difference of cubes formula is $(a - b)(a^{2}+ab + b^{2})=a^{3}-b^{3}$.

Step2: Analyze option A

For $(x + 7)(x^{2}-7x + 14)$, comparing with $(a + b)(a^{2}-ab + b^{2})$, here $b = 7$, but $b^{2}=49\neq14$, so it is not a sum or difference of cubes.

Step3: Analyze option B

For $(x + 8)(x^{2}+8x + 64)$, comparing with $(a + b)(a^{2}-ab + b^{2})$, the middle - term sign is wrong. It should be $- 8x$ for the sum - of - cubes formula, so it is not a sum or difference of cubes.

Step4: Analyze option C

For $(x - 9)(x^{2}+9x + 81)$, comparing with $(a - b)(a^{2}+ab + b^{2})$, where $a=x$ and $b = 9$, we have $(x - 9)(x^{2}+9x + 81)=x^{3}-9^{3}=x^{3}-729$, which is a difference of cubes.

Step5: Analyze option D

For $(x - 10)(x^{2}-10x + 100)$, comparing with $(a - b)(a^{2}+ab + b^{2})$, the middle - term sign is wrong. It should be $+10x$ for the difference - of - cubes formula, so it is not a sum or difference of cubes.