Answer
$(a^2b^2+4)(ab+2)(ab-2)$
Work Step by Step
Using $a^2-b^2=(a+b)(a-b)$ or the factoring of the difference of $2$ squares, then the factored form of the given expression, $
a^4b^4-16
$, is
\begin{array}{l}
(a^2b^2+4)(a^2b^2-4)
\\\\=
(a^2b^2+4)(ab+2)(ab-2)
.\end{array}