Factor the expression $18m^3 - 8mn^2$.

1. Problem Description

Factor the expression $18m^3 - 8mn^2$.

2. Solution Steps

First, we look for the greatest common factor (GCF) of the two terms. The GCF of the coefficients 18 and 8 is

2. The GCF of $m^3$ and $mn^2$ is $m$. So, the GCF of the two terms is $2m$.

We factor out the GCF:

$18m^3 - 8mn^2 = 2m(9m^2 - 4n^2)$.

Now we examine the expression inside the parenthesis, $9m^2 - 4n^2$. This is a difference of squares, which can be factored as follows:

$a^2 - b^2 = (a+b)(a-b)$

In our case, $9m^2 = (3m)^2$ and $4n^2 = (2n)^2$.

Therefore, $9m^2 - 4n^2 = (3m)^2 - (2n)^2 = (3m + 2n)(3m - 2n)$.

Substitute this back into the expression:

$2m(9m^2 - 4n^2) = 2m(3m + 2n)(3m - 2n)$.

3. Final Answer

$2m(3m+2n)(3m-2n)$