## Intermediate Algebra: Connecting Concepts through Application

$7(g^2+9h^2)(g+3h)(g-3h)$
$\bf{\text{Solution Outline:}}$ To factor the given expression, $7g^4-567h^4 ,$ factor first the $GCF.$ Then use the factoring of the sum or difference of $2$ squares. $\bf{\text{Solution Details:}}$ The $GCF$ of the terms is $GCF= 7$ since it is the highest number that can evenly divide (no remainder) all the given terms. Factoring the $GCF,$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7(g^4-81h^4) .\end{array} The expressions $g^4$ and $81h^4$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $g^4-81h^4 ,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7[(g^2)^2-(9h^2)^2] \\\\= 7(g^2+9h^2)(g^2-9h^2) .\end{array} The expressions $g^2$ and $9h^2$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $g^2-9h^2 ,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to \begin{array}{l}\require{cancel} 7(g^2+9h^2)[(g)^2-(3h)^2] \\\\= 7(g^2+9h^2)(g+3h)(g-3h) .\end{array}