## 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 $2{x}^{2}+5x+3$ can be rewritten as $\left(2x+3\right)\left(x+1\right)$ using this process. We begin by rewriting the original expression as $2{x}^{2}+2x+3x+3$ 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.

### A General Note: Factor by Grouping

To factor a trinomial in the form $a{x}^{2}+bx+c$ 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 $a{x}^{2}+bx+c$, 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 $a{x}^{2}+px+qx+c$.
4. Pull out the GCF of $a{x}^{2}+px$.
5. Pull out the GCF of $qx+c$.
6. Factor out the GCF of the expression.

### Example 3: Factoring a Trinomial by Grouping

Factor $5{x}^{2}+7x - 6$ by grouping.

### Solution

We have a trinomial with $a=5,b=7$, and $c=-6$. First, determine $ac=-30$. We need to find two numbers with a product of $-30$ and a sum of $7$. In the table, we list factors until we find a pair with the desired sum.

Factors of $-30$ Sum of Factors
$1,-30$ $-29$
$-1,30$ 29
$2,-15$ $-13$
$-2,15$ 13
$3,-10$ $-7$
$-3,10$ 7

So $p=-3$ and $q=10$.

$\begin{array}{cc}5{x}^{2}-3x+10x - 6 \hfill & \text{Rewrite the original expression as }a{x}^{2}+px+qx+c.\hfill \\ x\left(5x - 3\right)+2\left(5x - 3\right)\hfill & \text{Factor out the GCF of each part}.\hfill \\ \left(5x - 3\right)\left(x+2\right)\hfill & \text{Factor out the GCF}\text{ }\text{ of the expression}.\hfill \end{array}$

### Analysis of the Solution

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

### Try It 3

Factor the following.

a.$2{x}^{2}+9x+9$
b. $6{x}^{2}+x - 1$

Solution