Example 3 from Algebra and Trigonometry, 1.5 Factoring Polynomials

Factoring a Trinomial by Grouping

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

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We have a trinomial with $a = 5, b = 7,$ and $c = −6.$ First, determine $a c = −30.$ We need to find two numbers with a product of $−30$ and a sum of $7.$ In the table below, 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 $q = 10.$

$\begin{matrix} 5 x^{2} - 3 x + 10 x - 6 & Rewrite the original expression as a x^{2} + p x + q x + c . \\ x (5 x - 3) + 2 (5 x - 3) & Factor out the GCF of each part . \\ (5 x - 3) (x + 2) & Factor out the GCF of the expression . \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Factor

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

How did it go?