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