Answer
$(x^3-6)(x^3+2)$
Work Step by Step
Let $z=x^3$. Then the given expression, $
x^6-4x^3-12
$, is equivalent to $
z^2-4z-12
$.
The two numbers whose product is $ac=
1(-12)=-12
$ and whose sum is $b=
-4
$ are $\{
-6,2
\}$. Using these two numbers to decompose the middle term, then the factored form of the expression, $
z^2-4z-12
$, is
\begin{array}{l}\require{cancel}
z^2-6z+2z-12
\\\\=
(z^2-6z)+(2z-12)
\\\\=
z(z-6)+2(z-6)
\\\\=
(z-6)(z+2)
.\end{array}
Since $z=x^3$, then,
\begin{array}{l}
(z-6)(z+2)
\\\\=
(x^3-6)(x^3+2)
.\end{array}
