Answer
$(2mn-1)(4mn-3)$
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, $
8m^2n^2-10mn+3
,$ the value of $ac$ is $
8(3)=24
$ and the value of $b$ is $
-10
.$
The possible pairs of integers whose product is $ac$ are
\begin{array}{l}\require{cancel}
\{1,24\}, \{2,12\}, \{3,8\}, \{4,6\},
\{-1,-24\}, \{-2,-12\}, \{-3,-8\}, \{-4,-6\}
.\end{array}
Among these pairs, the one that gives a sum of $b$ is $\{
-4,-6
\}.$ Using these $2$ numbers to decompose the middle term of the given expression results to
\begin{array}{l}\require{cancel}
8m^2n^2-4mn-6mn+3
.\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}
(8m^2n^2-4mn)-(6mn-3)
.\end{array}
Factoring the $GCF$ in each group results to
\begin{array}{l}\require{cancel}
4mn(2mn-1)-3(2mn-1)
.\end{array}
Factoring the $GCF=
(2mn-1)
$ of the entire expression above results to
\begin{array}{l}\require{cancel}
(2mn-1)(4mn-3)
.\end{array}
