###### Example144

Factor \(x^2+12x+20\text{.}\)

Determine the factors of the last term whose sum equals the coefficient of the middle term. To do this, list all of the factorizations of \(20\) and search for factors whose sum equals \(12\text{.}\)

\(1\cdot 20\) | \(1 +20=21\) |

\(2\cdot 10\) | \(\alert{2 +10=12}\) |

\(4\cdot 5\) | \(4 +5=9\) |

Choose \(20 = 2 \cdot 10\) because \(2 + 10 = 12\text{.}\) Therefore, \(12x=2x+10x\text{,}\) and we can write

We factor the equivalent expression by grouping.

Our factored form is \((x+10)(x+2)\text{.}\)

We check by multiplying the two binomials.

Note: We made a choice in the above problem to write \(12x=2x+10x\text{.}\) What would happen if we instead had written \(12x=10x+2x\text{?}\) Notice that in the following calculation the answer is equivalent.

We factor the equivalent expression by grouping.