Answer
$(3a-2b)(5a-4b)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the quadratic expression $ax^2+bx+c,$ find two numbers whose product is $ac$ and whose sum is $b$. Use these $2$ numbers to decompose the middle term of the quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
In the given expression, $
15a^2-22ab+8b^2
,$ the value of $ac$ is $
15(8)=120
$ and the value of $b$ is $
-22
.$ The $2$ numbers that have a product $ac$ and a sum of $b$ are $\{
-10,-12
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
15a^2-10ab-12ab+8b^2
.\end{array}
Grouping the first and third terms and the second and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(15a^2-10ab)-(12ab-8b^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
5a(3a-2b)-4b(3a-2b)
.\end{array}
Factoring the $GCF=
(3a-2b)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(3a-2b)(5a-4b)
.\end{array}
