## Intermediate Algebra: Connecting Concepts through Application

$4x(x-3)(5x+2)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $20x^3-52x^2-24x ,$ factor first the $GCF.$ Then 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. $\bf{\text{Solution Details:}}$ Factoring the $GCF= 4x ,$ the given expression is equivalent to \begin{array}{l}\require{cancel} 4x(5x^2-13x-6) .\end{array} Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $5(-6)=-30$ and the value of $b$ is $-13 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ -15,2 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 4x(5x^2-15x+2x-6) .\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} 4x[(5x^2-15x)+(2x-6)] .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 4x[5x(x-3)+2(x-3)] .\end{array} Factoring the $GCF= (x-3)$ of the entire expression above results to \begin{array}{l}\require{cancel} 4x[(x-3)(5x+2)] \\\\= 4x(x-3)(5x+2) .\end{array}