The problem asks us to factor the expression $9x^3 + 30x^2 - 24x$ by first factoring out the greatest common factor (GCF).

1. Problem Description

The problem asks us to factor the expression $9x^3 + 30x^2 - 24x$ by first factoring out the greatest common factor (GCF).

2. Solution Steps

First, we need to identify the greatest common factor of the coefficients and the variables.

The coefficients are 9, 30, and -

2

4. We need to find the greatest common divisor (GCD) of these numbers.

The factors of 9 are 1, 3, and

9. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, and

3

0. The factors of -24 are 1, 2, 3, 4, 6, 8, 12, and

2

4. The greatest common factor of 9, 30, and -24 is

3.

Now let's look at the variables. We have $x^3$, $x^2$, and $x$.

The greatest common factor of $x^3$, $x^2$, and $x$ is $x$.

Therefore, the greatest common factor of $9x^3$, $30x^2$, and $-24x$ is $3x$.

Now we factor out $3x$ from the expression:

$9x^3 + 30x^2 - 24x = 3x(3x^2 + 10x - 8)$

Now we need to factor the quadratic expression $3x^2 + 10x - 8$. We look for two numbers that multiply to $3 \cdot (-8) = -24$ and add to

1

0. These numbers are 12 and -

2. So we rewrite the middle term:

$3x^2 + 10x - 8 = 3x^2 + 12x - 2x - 8$

Now we factor by grouping:

$3x^2 + 12x - 2x - 8 = 3x(x + 4) - 2(x + 4) = (3x - 2)(x + 4)$

Thus, $3x^2 + 10x - 8 = (3x - 2)(x + 4)$.

Therefore, $9x^3 + 30x^2 - 24x = 3x(3x^2 + 10x - 8) = 3x(3x - 2)(x + 4)$.

3. Final Answer

$3x(3x-2)(x+4)$