Answer
$a^2b^2-1$
Work Step by Step
Using $(a+b)(a-b)=a^2-b^2$ or the product of the sum and difference of like terms, the product of the given expression, $
(ab+1)(ab-1)
,$ is
\begin{array}{l}\require{cancel}
(ab)^2-(1)^2
\\\\=
a^2b^2-1
.\end{array}