Simplify: $\;$ $\left[\left(a - b\right) \sqrt{\dfrac{a + b}{a - b}} + a - b\right] \left(a - b\right) \left(\sqrt{\dfrac{a + b}{a - b}} - 1\right)$, $\;\;$ $a + b < 0$, $\;$ $a - b < 0$
$\left[\left(a - b\right) \sqrt{\dfrac{a + b}{a - b}} + a - b\right] \left(a - b\right) \left(\sqrt{\dfrac{a + b}{a - b}} - 1\right)$
$= \left[\dfrac{\left(a - b\right) \sqrt{a + b} + \left(a - b\right) \sqrt{a - b}}{\sqrt{a - b}}\right] \left(a - b\right) \left(\dfrac{\sqrt{a + b} - \sqrt{a - b}}{\sqrt{a - b}}\right)$
$= \dfrac{\left(a - b\right) \left(\sqrt{a + b} + \sqrt{a - b}\right)}{a - b} \times \left(a - b\right) \times \left(\sqrt{a + b} - \sqrt{a - b}\right)$
$= \left(a - b\right) \left(a + b - a + b\right)$
$= 2b \left(a - b\right)$