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