Answer
$(5m-2n)(m-4n)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
5m^2-22mn+8n^2
,$ find two numbers whose product is $ac$ and whose sum is $b$ in the quadratic expression $ax^2+bx+c.$ Use these $2$ numbers to decompose the middle term of the given quadratic expression and then use factoring by grouping.
$\bf{\text{Solution Details:}}$
Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $
5(8)=40
$ and the value of $b$ is $
-22
.$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{
-2,-20
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{array}{l}\require{cancel}
5m^2-2mn-20mn+8n^2
.\end{array}
Grouping the first and second terms and the third and fourth terms, the given expression is equivalent to
\begin{array}{l}\require{cancel}
(5m^2-2mn)-(20mn-8n^2)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
m(5m-2n)-4n(5m-2n)
.\end{array}
Factoring the $GCF=
(5m-2n)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(5m-2n)(m-4n)
.\end{array}