## Intermediate Algebra (12th Edition)

$(2mn-1)(4mn-3)$
$\bf{\text{Solution Outline:}}$ 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}