# 4.4 Factoring by Grouping

Trinomials with leading coefficients other than 1 are slightly more complicated to factor. For these trinomials, we can factor by grouping by dividing the $x$ term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression. The trinomial $2x^2+5x+3$ can be rewritten as $\left(2x+3\right)\left(x+1\right)$ using this process. We begin by rewriting the original expression as $2x^2+2x+3x+3$ and then factor each portion of the expression to obtain $2x\left(x+1\right)+3\left(x+1\right)$. We then pull out the GCF of $\left(x+1\right)$ to find the factored expression.

## Factor by Grouping

To factor a trinomial in the form $ax^2+bx+c$ by grouping, we find two numbers with a product of $ac$ and a sum of $b$. We use these numbers to divide the $x$ term into the sum of two terms and factor each portion of the expression separately, then factor out the GCF of the entire expression.

## How To

Given a trinomial in the form $ax^2+bx+c$, factor by grouping.

1. List factors of $ac$.
2. Find $p$ and $q$, a pair of factors of $ac$ with a sum of $b$.
3. Rewrite the original expression as $ax^2+px+qx+c$.
4. Pull out the GCF of $ax^2+px$.
5. Pull out the GCF of $qx+c$.
6. Factor out the GCF of the expression.

## Example 1: Factoring a Trinomial by Grouping

1. Factor $5x^2+7x−6$ by grouping.

### Analysis

We can check our work by multiplying. Use FOIL to confirm that $\left(5x-3\right)\left(x+2\right)=5x^2+7x-6$.

1. Factor the following:
1. $2x^2+9x+9$
2. $6x^2+x− 1$