Question

which expression is equivalent to $x^3 - x^2 - 14x + 24$?
a $(x - 2)(x - 3)(x + 4)$
b $(x - 2)(x + 3)(x - 4)$
c $(x + 2)(x - 3)(x - 4)$
d $(x + 2)(x + 3)(x + 4)$

Explanation:

Step1: Expand Option A

First, multiply \((x - 2)(x - 3)\):
\((x - 2)(x - 3)=x^2 - 3x - 2x + 6=x^2 - 5x + 6\).
Then multiply by \((x + 4)\):
\((x^2 - 5x + 6)(x + 4)=x^3 + 4x^2 - 5x^2 - 20x + 6x + 24=x^3 - x^2 - 14x + 24\).

Step2: Verify Other Options (Optional)

For Option B: \((x - 2)(x + 3)=x^2 + 3x - 2x - 6=x^2 + x - 6\); \((x^2 + x - 6)(x - 4)=x^3 - 4x^2 + x^2 - 4x - 6x + 24=x^3 - 3x^2 - 10x + 24\) (not matching).
For Option C: \((x + 2)(x - 3)=x^2 - 3x + 2x - 6=x^2 - x - 6\); \((x^2 - x - 6)(x - 4)=x^3 - 4x^2 - x^2 + 4x - 6x + 24=x^3 - 5x^2 - 2x + 24\) (not matching).
For Option D: \((x + 2)(x + 3)=x^2 + 3x + 2x + 6=x^2 + 5x + 6\); \((x^2 + 5x + 6)(x + 4)=x^3 + 4x^2 + 5x^2 + 20x + 6x + 24=x^3 + 9x^2 + 26x + 24\) (not matching).

Answer:

A. \((x - 2)(x - 3)(x + 4)\)