The problem is to factor the given expression $9m^5n^2 + 12m^3n$.

1. Problem Description

The problem is to factor the given expression $9m^5n^2 + 12m^3n$.

2. Solution Steps

To factor the expression $9m^5n^2 + 12m^3n$, we need to find the greatest common factor (GCF) of the two terms.

First, find the GCF of the coefficients 9 and

1

2. The factors of 9 are 1, 3, and

9. The factors of 12 are 1, 2, 3, 4, 6, and

1

2. The GCF of 9 and 12 is

3.

Next, find the GCF of the variable parts.

The first term has $m^5$ and the second term has $m^3$. The GCF of $m^5$ and $m^3$ is $m^3$.

The first term has $n^2$ and the second term has $n$. The GCF of $n^2$ and $n$ is $n$.

Therefore, the GCF of the two terms is $3m^3n$.

Now, divide each term by the GCF:

$\frac{9m^5n^2}{3m^3n} = 3m^{5-3}n^{2-1} = 3m^2n$

$\frac{12m^3n}{3m^3n} = 4$

So, we can write the expression as:

$9m^5n^2 + 12m^3n = 3m^3n(3m^2n + 4)$.

3. Final Answer

The factored expression is $3m^3n(3m^2n + 4)$.