Answer
$(a^2-8)(a^2+6)$
Work Step by Step
Let $z=a^2.$ Then the given expression, $
a^4-2a^2-48
,$ is equivalent to
\begin{array}{l}\require{cancel}
z^2-2z-48
.\end{array}
The two numbers whose product is $ac=
1(-48)=-48
$ and whose sum is $b=
-2
$ are $\{
-8,6
\}.$ Using these two numbers to decompose the middle term, then the factored form of $
z^2-2z-48
$ is
\begin{array}{l}\require{cancel}
z^2-8z+6z-48
\\\\=
(z^2-8z)+(6z-48)
\\\\=
z(z-8)+6(z-8)
\\\\=
(z-8)(z+6)
.\end{array}
Since $z=a^2$, then the expression $
(z-8)(z+6)
$ is equivalent to
\begin{array}{l}\require{cancel}
(a^2-8)(a^2+6)
.\end{array}
