Answer
$(9p-5r)(2p+7r)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}