Answer
$\dfrac{4a^2}{9(a-1)}$
Work Step by Step
The given expression, $
\left( \dfrac{2a}{3} \right)^2\div\left( \dfrac{a^2}{a+1}-\dfrac{1}{a+1} \right)
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{4a^2}{9}\div\dfrac{a^2-1}{a+1}
\\\\
\dfrac{4a^2}{9}\div\dfrac{(a+1)(a-1)}{a+1}
\\\\
\dfrac{4a^2}{9}\cdot\dfrac{a+1}{(a+1)(a-1)}
\\\\
\dfrac{4a^2}{9}\cdot\dfrac{\cancel{a+1}}{(\cancel{a+1})(a-1)}
\\\\
\dfrac{4a^2}{9(a-1)}
.\end{array}
