Simplify: $\;$ $\left[\dfrac{a^{\frac{1}{2}} + 2}{a + 2 a ^{\frac{1}{2}} + 1} - \dfrac{a^{\frac{1}{2}} - 2}{a - 1}\right] \times \left[\dfrac{a^{\frac{1}{2}} + 1}{a^{\frac{1}{2}}}\right]$
$\left[\dfrac{a^{\frac{1}{2}} + 2}{a + 2 a ^{\frac{1}{2}} + 1} - \dfrac{a^{\frac{1}{2}} - 2}{a - 1}\right] \times \left[\dfrac{a^{\frac{1}{2}} + 1}{a^{\frac{1}{2}}}\right]$
$= \left[\dfrac{\sqrt{a} + 2}{a + 2 \sqrt{a} + 1} - \dfrac{\sqrt{a} - 2}{a - 1}\right] \times \left[\dfrac{\sqrt{a} + 1}{\sqrt{a}}\right]$
$= \left[\dfrac{\left(\sqrt{a} + 2\right) \left(a - 1\right) - \left(\sqrt{a} - 2\right) \left(a + 2 \sqrt{a} + 1\right)}{\left(\sqrt{a} + 1\right)^2 \left(a - 1\right)}\right] \times \left[\dfrac{\sqrt{a} + 1}{\sqrt{a}}\right]$
$= \dfrac{a \sqrt{a} - \sqrt{a} + 2a - 2 - \left(a \sqrt{a} + 2a + \sqrt{a} - 2a - 4 \sqrt{a} - 2\right)}{\sqrt{a} \left(\sqrt{a} + 1\right) \left(a - 1\right)}$
$= \dfrac{2a + 2 \sqrt{a}}{\sqrt{a} \left(\sqrt{a} + 1\right) \left(a - 1\right)}$
$= \dfrac{2 \sqrt{a} \left(\sqrt{a} + 1\right)}{\sqrt{a} \left(\sqrt{a} + 1\right) \left(a - 1\right)}$
$= \dfrac{2}{a - 1}$