## Intermediate Algebra (12th Edition)

$(9p-5r)(2p+7r)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $18p^2+53pr-35r^2 ,$ use the factoring of trinomials in the form $ax^2+bx+c.$ $\bf{\text{Solution Details:}}$ In the trinomial expression above, $a= 18 ,b= 53 ,\text{ and } c= -35 .$ Using the factoring of trinomials in the form $ax^2+bx+c,$ the two numbers whose product is $ac= 18(-35)=630$ and whose sum is $b$ are $\left\{ -10,63 \right\}.$ Using these two numbers to decompose the middle term results to \begin{array}{l}\require{cancel} 18p^2-10pr+63pr-35r^2 .\end{array} Using factoring by grouping, the expression above is equivalent to \begin{array}{l}\require{cancel} (18p^2-10pr)+(63pr-35r^2) \\\\= 2p(9p-5r)+7r(9p-5r) \\\\= (9p-5r)(2p+7r) .\end{array}