Simplify: $\;$ $\left(\dfrac{a \sqrt{a} + b \sqrt{b}}{\sqrt{a} + \sqrt{b}}\right) : \left(a - b\right) + \dfrac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}}$
$\left(\dfrac{a \sqrt{a} + b \sqrt{b}}{\sqrt{a} + \sqrt{b}}\right) : \left(a - b\right) + \dfrac{2 \sqrt{b}}{\sqrt{a} + \sqrt{b}}$
$= \dfrac{\left(a \sqrt{a} + b \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)} \times \dfrac{1}{a - b} + \dfrac{2 \sqrt{b} \left(\sqrt{a} - \sqrt{b}\right)}{\left(\sqrt{a} + \sqrt{b}\right) \left(\sqrt{a} - \sqrt{b}\right)}$
$= \dfrac{\left(a^2 - a \sqrt{ab} + b \sqrt{ab} - b^2\right)}{a - b} \times \dfrac{1}{a - b} + \dfrac{2 \sqrt{b} \left(\sqrt{a} - \sqrt{b}\right)}{a - b}$
$= \dfrac{\left(a + b\right) \left(a - b\right) - \sqrt{ab} \left(a - b\right)}{\left(a - b\right)^2} + \dfrac{2 \sqrt{b} \left(\sqrt{a} - \sqrt{b}\right)}{den}$
$= \dfrac{\left(a - b\right) \left(a + b - \sqrt{ab}\right)}{\left(a - b\right)^2} + \dfrac{2 \sqrt{ab} - 2b}{a - b}$
$= \dfrac{a + b - \sqrt{ab}}{a - b} + \dfrac{2 \sqrt{ab} - 2b}{a - b}$
$= \dfrac{a - b + \sqrt{ab}}{a - b}$
$= 1 + \dfrac{\sqrt{ab}}{a - b}$