Answer
$(x^3-3)(x^3-4)$
Work Step by Step
Let $z=x^3$. Then the given expression, $
x^6-7x^3+12
$, is equivalent to $
z^2-7z+12
$.
The two numbers whose product is $ac=
1(12)=12
$ and whose sum is $b=
-7
$ are $\{
-3,-4
\}$. Using these two numbers to decompose the middle term, then the factored form of the expression, $
z^2-7z+12
$, is
\begin{array}{l}\require{cancel}
z^2-3z-4z+12
\\\\=
(z^2-3z)-(4z-12)
\\\\=
z(z-3)-4(z-3)
\\\\=
(z-3)(z-4)
.\end{array}
Since $z=x^3$, then,
\begin{array}{l}
(z-3)(z-4)
\\\\=
(x^3-3)(x^3-4)
.\end{array}
