Answer
$7(g^2+9h^2)(g+3h)(g-3h)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
