The problem asks us to factor the expression $8a^3b - 18ab^3$ completely. If the expression is prime, we should indicate that.

1. Problem Description

The problem asks us to factor the expression $8a^3b - 18ab^3$ completely. If the expression is prime, we should indicate that.

2. Solution Steps

First, we identify the greatest common factor (GCF) of the two terms in the expression $8a^3b - 18ab^3$.

The numerical coefficients are 8 and

1

8. Their greatest common factor is

2. The variable factors are $a^3b$ and $ab^3$. Their greatest common factor is $ab$.

Therefore, the GCF of $8a^3b$ and $-18ab^3$ is $2ab$.

We can factor out $2ab$ from the expression:

$8a^3b - 18ab^3 = 2ab(4a^2 - 9b^2)$

Now we look at the expression inside the parenthesis: $4a^2 - 9b^2$. This is a difference of two squares.

We can use the difference of squares factorization:

$x^2 - y^2 = (x+y)(x-y)$

In our case, $4a^2 = (2a)^2$ and $9b^2 = (3b)^2$, so $x = 2a$ and $y = 3b$.

Therefore, $4a^2 - 9b^2 = (2a + 3b)(2a - 3b)$.

So, the complete factorization is:

$8a^3b - 18ab^3 = 2ab(4a^2 - 9b^2) = 2ab(2a + 3b)(2a - 3b)$

3. Final Answer

$2ab(2a + 3b)(2a - 3b)$