## Intermediate Algebra: Connecting Concepts through Application

$(5m-2n)(m-4n)$
$\bf{\text{Solution Outline:}}$ 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}