## Intermediate Algebra (12th Edition)

$(4a+5b)^2$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $16a^2+40ab+25b^2 ,$ 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 $16(25)=400$ and the value of $b$ is $40 .$ The $2$ numbers that have a product of $ac$ and a sum of $b$ are $\left\{ 20,20 \right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to \begin{array}{l}\require{cancel} 16a^2+20ab+20ab+25b^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} (16a^2+20ab)+(20ab+25b^2) .\end{array} Factoring the $GCF$ in each group results to \begin{array}{l}\require{cancel} 4a(4a+5b)+5b(4a+5b) .\end{array} Factoring the $GCF= (4a+5b)$ of the entire expression above results to \begin{array}{l}\require{cancel} (4a+5b)(4a+5b) \\\\= (4a+5b)^2 .\end{array}