## Intermediate Algebra: Connecting Concepts through Application

$(3x+5)^2$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $9x^2+30x+25 ,$ 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 $9(25)=225$ and the value of $b$ is $30 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ 15,15 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 9x^2+15x+15x+25 .\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} (9x^2+15x)+(15x+25) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 3x(3x+5)+5(3x+5) .\end{array} Factoring the $GCF= (3x+5)$ of the entire expression above results to \begin{array}{l}\require{cancel} (3x+5)(3x+5) \\\\= (3x+5)^2 .\end{array}