## Intermediate Algebra: Connecting Concepts through Application

$(m-3)(7m-4)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $7m^2-25m+12 ,$ 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 $7(12)=84$ and the value of $b$ is $-25 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -21,-4 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 7m^2-21m-4m+12 .\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} (7m^2-21m)-(4m-12) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 7m(m-3)-4(m-3) .\end{array} Factoring the $GCF= (m-3)$ of the entire expression above results to \begin{array}{l}\require{cancel} (m-3)(7m-4) .\end{array}