Answer
$\dfrac{2a}{a+2}$
Work Step by Step
Factoring the expressions and then cancelling the common factor/s between the numerator and the denominator, then the given expression, $
\dfrac{6a^2b^2}{a^2-4}\div\dfrac{3ab^2}{a-2}
$, simplifies to
\begin{array}{l}\require{cancel}
\dfrac{6a^2b^2}{a^2-4}\cdot\dfrac{a-2}{3ab^2}
\\\\=
\dfrac{3ab^2\cdot2a}{(a+2)(a-2)}\cdot\dfrac{a-2}{3ab^2}
\\\\=
\dfrac{\cancel{3ab^2}\cdot2a}{(a+2)(\cancel{a-2})}\cdot\dfrac{\cancel{a-2}}{\cancel{3ab^2}}
\\\\=
\dfrac{2a}{a+2}
.\end{array}
