The problem asks us to factor the expression $4x^3 + 14x^2 - 30x$ by first factoring out the greatest common factor (GCF).

1. Problem Description

The problem asks us to factor the expression $4x^3 + 14x^2 - 30x$ by first factoring out the greatest common factor (GCF).

2. Solution Steps

First, we need to find the GCF of the coefficients 4, 14, and -

3

0. The factors of 4 are 1, 2, and

4. The factors of 14 are 1, 2, 7, and

1

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

3

0. The GCF of 4, 14, and 30 is

2.

Next, we look for the GCF of the variable terms $x^3$, $x^2$, and $x$. The GCF is $x$.

Therefore, the GCF of the entire expression is $2x$.

Now, we factor out $2x$ from each term:

$4x^3 = 2x * 2x^2$

$14x^2 = 2x * 7x$

$-30x = 2x * -15$

So, we have:

$4x^3 + 14x^2 - 30x = 2x(2x^2 + 7x - 15)$

Now, we try to factor the quadratic expression $2x^2 + 7x - 15$.

We look for two numbers that multiply to $2*(-15) = -30$ and add to

7. Those numbers are 10 and -

3. So we can rewrite the middle term as $7x = 10x - 3x$.

Then, we have $2x^2 + 10x - 3x - 15$.

Now we factor by grouping:

$2x^2 + 10x = 2x(x+5)$

$-3x - 15 = -3(x+5)$

So, $2x^2 + 10x - 3x - 15 = 2x(x+5) - 3(x+5) = (2x-3)(x+5)$

Therefore, $2x^2 + 7x - 15 = (2x-3)(x+5)$.

Substituting this back into the original expression, we have:

$4x^3 + 14x^2 - 30x = 2x(2x-3)(x+5)$

3. Final Answer

$2x(2x-3)(x+5)$