Factorize each of the following expression:

$a^{4} b^{4}-81 c^{4}$

$=\left(a^{2} b^{2}\right)^{2}-\left(9 c^{2}\right)^{2}$

$=\left(a^{2} b^{2}+9 c^{2}\right)\left(a^{2} b^{2}-9 c^{2}\right)$

$=\left(a^{2} b^{2}+9 c^{2}\right)\left[(a b)^{2}-(3 c)^{2}\right]$

$=\left(a^{2} b^{2}+9 c^{2}\right)(a b+3 c)(a b-3 c)$