The problem asks us to factor the polynomial $18ax^2 + 48ax - 18a$ completely.

1. Problem Description

The problem asks us to factor the polynomial $18ax^2 + 48ax - 18a$ completely.

2. Solution Steps

First, we look for the greatest common factor (GCF) of the terms in the polynomial.

The coefficients are 18, 48, and -

1

8. The GCF of the coefficients is

6. All the terms have $a$, so $a$ is part of the GCF.

Therefore, the GCF is $6a$.

Factoring out $6a$ from the polynomial, we get

$18ax^2 + 48ax - 18a = 6a(3x^2 + 8x - 3)$.

Now, we need to factor the quadratic $3x^2 + 8x - 3$. We are looking for two numbers that multiply to $3*(-3) = -9$ and add up to

8. These numbers are 9 and -

1. We rewrite the middle term $8x$ as $9x - x$:

$3x^2 + 8x - 3 = 3x^2 + 9x - x - 3$.

Now, we factor by grouping:

$3x^2 + 9x - x - 3 = 3x(x + 3) - 1(x + 3) = (3x - 1)(x + 3)$.

Thus, $3x^2 + 8x - 3 = (3x - 1)(x + 3)$.

Therefore, the factored form of the original polynomial is $6a(3x - 1)(x + 3)$.

3. Final Answer

$6a(3x - 1)(x + 3)$