Simplify: $\;$ $\left[\dfrac{x^3 + y^3}{x + y} : \left(x^2 - y^2\right)\right]+ \left[\dfrac{2y}{x + y} - \dfrac{xy}{x^2 - y^2}\right]$
$\left[\dfrac{x^3 + y^3}{x + y} : \left(x^2 - y^2\right)\right]+ \left[\dfrac{2y}{x + y} - \dfrac{xy}{x^2 - y^2}\right]$
$= \left[\dfrac{\left(x + y\right) \left(x^2 - xy + y^2\right)}{x + y} : \left(x^2 - y^2\right)\right] + \left[\dfrac{2y}{x + y} - \dfrac{xy}{\left(x + y\right) \left(x - y\right)}\right]$
$= \dfrac{x^2 - xy + y^2}{x^2 - y^2} + \dfrac{1}{x + y} \left[2y - \dfrac{xy}{x - y}\right]$
$= \dfrac{x^2 - xy + y^2}{x^2 - y^2} + \dfrac{2xy - 2y^2 - xy}{\left(x + y\right) \left(x - y\right)}$
$= \dfrac{x^2 - xy + y^2}{x^2 - y^2} + \dfrac{xy - 2y^2}{x^2 - y^2}$
$= \dfrac{x^2 - xy + y^2 + xy - 2y^2}{x^2 - y^2}$
$= \dfrac{x^2 - y^2}{x^2 - y^2}$
$= 1$