Simplify: $\;$ $\left(\dfrac{1 - x^{-2}}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}} - \dfrac{2}{x^2 : \sqrt{x}} + \dfrac{x^{-2} - x}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}}\right) \left(1 + \dfrac{2}{x^2}\right)^{-1}$
$\left(\dfrac{1 - x^{-2}}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}} - \dfrac{2}{x^2 : \sqrt{x}} + \dfrac{x^{-2} - x}{x^{\frac{1}{2}} - x^{\frac{-1}{2}}}\right) \left(1 + \dfrac{2}{x^2}\right)^{-1}$
$= \left(\dfrac{1 - \dfrac{1}{x^2}}{\sqrt{x} - \dfrac{1}{\sqrt{x}}} - \dfrac{2 \sqrt{x}}{x^2} + \dfrac{\dfrac{1}{x^2} - x}{\sqrt{x} - \dfrac{1}{\sqrt{x}}}\right) \left(\dfrac{x^2 + 2}{x^2}\right)^{-1}$
$= \left(\dfrac{\dfrac{x^2 - 1}{x^2}}{\dfrac{x - 1}{\sqrt{x}}} - \dfrac{2 \sqrt{x}}{x^2} + \dfrac{\dfrac{1 - x^3}{x^2}}{\dfrac{x - 1}{\sqrt{x}}}\right) \left(\dfrac{x^2}{x^2 + 2}\right)$
$= \left(\dfrac{\left(x + 1\right) \left(x - 1\right) \sqrt{x}}{x^2 \left(x - 1\right)} - \dfrac{2 \sqrt{x}}{x^2} + \dfrac{\sqrt{x} \left(1 - x\right) \left(1 + x + x^2\right)}{x^2 \left(x - 1\right)}\right) \left(\dfrac{x^2}{x^2 + 2}\right)$
$= \left(\sqrt{x} \left(x + 1\right) - 2 \sqrt{x} - \sqrt{x} \left(1 + x + x^2\right)\right) \times \dfrac{1}{x^2 + 2}$
$= \dfrac{x \sqrt{x} + \sqrt{x} - 2 \sqrt{x} - \sqrt{x} - x \sqrt{x} - x^2 \sqrt{x}}{x^2 + 2}$
$= \dfrac{- 2 \sqrt{x} - x^2 \sqrt{x}}{x^2 + 2}$
$= \dfrac{- \sqrt{x} \left(x^2 + 2\right)}{x^2 + 2}$
$= - \sqrt{x}$