The problem asks us to factor the expression $12a^3b - 27ab^3$.

1. Problem Description

The problem asks us to factor the expression $12a^3b - 27ab^3$.

2. Solution Steps

First, we find the greatest common factor (GCF) of the two terms $12a^3b$ and $27ab^3$.

The GCF of 12 and 27 is

3. The GCF of $a^3$ and $a$ is $a$. The GCF of $b$ and $b^3$ is $b$.

Therefore, the GCF of the two terms is $3ab$.

We can factor out $3ab$ from the expression:

$12a^3b - 27ab^3 = 3ab(4a^2 - 9b^2)$

Now we notice that $4a^2 - 9b^2$ is a difference of squares: $4a^2 = (2a)^2$ and $9b^2 = (3b)^2$.

We can use the difference of squares formula:

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

In this case, $x = 2a$ and $y = 3b$.

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

Thus, the fully factored expression is:

$12a^3b - 27ab^3 = 3ab(2a - 3b)(2a + 3b)$

3. Final Answer

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