Simplify: $\;$ $\left[\left(\dfrac{y}{y - x}\right)^{-2} - \dfrac{\left(x + y\right)^2 - 4xy}{x^2 - xy}\right] \dfrac{x^4}{x^2 y^2 - y^4}$
$\left[\left(\dfrac{y}{y - x}\right)^{-2} - \dfrac{\left(x + y\right)^2 - 4xy}{x^2 - xy}\right] \dfrac{x^4}{x^2 y^2 - y^4}$
$= \left[\left(\dfrac{y - x}{y}\right)^2 - \dfrac{x^2 + y^2 + 2xy - 4xy}{x \left(x - y\right)}\right] \dfrac{x^4}{y^2 \left(x^2 - y^2\right)}$
$= \left[\dfrac{\left(y - x\right)^2}{y^2} - \dfrac{x^2 + y^2 - 2xy}{x \left(x - y\right)}\right] \dfrac{x^4}{y^2 \left(x + y\right) \left(x - y\right)}$
$= \left[\dfrac{\left(y - x\right)^2}{y^2} - \dfrac{\left(y - x\right)^2}{x \left(x - y\right)}\right] \dfrac{x^4}{y^2 \left(x + y\right) \left(x - y\right)}$
$= \left(y - x\right)^2 \left[\dfrac{1}{y^2} - \dfrac{1}{x \left(x - y\right)}\right] \dfrac{x^4}{y^2 \left(x + y\right) \left(x - y\right)}$
$= \dfrac{\left(y - x\right)^2 \left(x^2 - xy - y^2\right) x^4}{y^2 x \left(x - y\right) y^2 \left(x + y\right) \left(x - y\right)}$
$= \dfrac{\left(y - x\right)^2 \left(x^2 - xy - y^2\right) x^3}{y^4 \left(x + y\right) \left(x - y\right)^2}$
$= \dfrac{x^3 \left(x^2 - xy - y^2\right)}{y^4 \left(x + y\right)}$