## Intermediate Algebra: Connecting Concepts through Application

$2b(3a+7)^2$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $18a^2b+84ab+98b ,$ 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 $GCF= 2b ,$ the given expression is equivalent to \begin{array}{l}\require{cancel} 2b(9a^2+42a+49) .\end{array} Using factoring of trinomials, the value of $ac$ in the trinomial expression above is $9(49)=441$ and the value of $b$ is $42 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ 21,21 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 2b(9a^2+21a+21a+49) .\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} 2b[(9a^2+21a)+(21a+49)] .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 2b[3a(3a+7)+7(3a+7)] .\end{array} Factoring the $GCF= (3a+7)$ of the entire expression above results to \begin{array}{l}\require{cancel} 2b[(3a+7)(3a+7)] \\\\= 2b(3a+7)^2 .\end{array}