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