Simplify: $\;$ $\left[\dfrac{a^2}{a + b} - \dfrac{a^3}{a^2 + 2ab + b^2}\right] : \left[\dfrac{a}{a + b} - \dfrac{a^2}{a^2 - b^2}\right]$ $\;$ and calculate it for $\;$ $a = - 2.5$, $\;$ $b = 0.5$.
$\left[\dfrac{a^2}{a + b} - \dfrac{a^3}{a^2 + 2ab + b^2}\right] : \left[\dfrac{a}{a + b} - \dfrac{a^2}{a^2 - b^2}\right]$
$= \left[\dfrac{a^2}{a + b} - \dfrac{a^3}{\left(a + b\right)^2}\right] : \left[\dfrac{a}{a + b} - \dfrac{a^2}{\left(a + b\right) \left(a - b\right)}\right]$
$= \left[\dfrac{a^2}{a + b} \left(1 - \dfrac{a}{a + b}\right)\right] : \left[\dfrac{a}{a + b} \left(1 - \dfrac{a}{a - b}\right)\right]$
$= \left[\dfrac{a^2 b}{\left(a + b\right)^2}\right] : \left[\dfrac{-ab}{\left(a + b\right) \left(a - b\right)}\right]$
$= \dfrac{a^2 b}{\left(a + b\right)^2} \times \dfrac{\left(a + b\right) \left(a - b\right)}{\left(-ab\right)}$
$= \dfrac{a \left(b - a\right)}{a + b}$
When $\;$ $a = -2.5$, $\;$ $b = 0.5$, $\;$ the given expression becomes
$\dfrac{-2.5 \times \left(0.5 + 2.5\right)}{-2.5 + 0.5}$
$= \dfrac{-2.5 \times 3}{-2}$
$= \dfrac{7.5}{2} = \dfrac{15}{4}$