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 and the variable terms.

The coefficients are 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 greatest common factor of 4, 14, and 30 is

2. The variable terms are $x^3$, $x^2$, and $x$. The lowest power of $x$ is $x^1$ or $x$.

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

Next, we factor out the GCF from each term:

$4x^3 = 2x(2x^2)$

$14x^2 = 2x(7x)$

$-30x = 2x(-15)$

So we can rewrite the original expression as:

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

Now we need to factor the quadratic expression $2x^2 + 7x - 15$. We are looking for two numbers that multiply to $2*(-15) = -30$ and add up to

7. The two numbers are 10 and -

3. So we can rewrite the quadratic as:

$2x^2 + 7x - 15 = 2x^2 + 10x - 3x - 15$

Factor by grouping:

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

Finally, we substitute this back into the expression with the GCF:

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

3. Final Answer

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