Simplify: $\;$ $\dfrac{2}{a} - \left\{\left[\dfrac{a + 1}{a^3 - 1} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a} \right] : \dfrac{a^3 + a^2 + 2a}{a^3 - 1} \right\}$
$\dfrac{2}{a} - \left\{\left[\dfrac{a + 1}{a^3 - 1} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a} \right] : \dfrac{a^3 + a^2 + 2a}{a^3 - 1} \right\}$ $\;\;\; \cdots \; (1)$
Consider the expression
$\left[\dfrac{a + 1}{a^3 - 1} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a} \right] : \dfrac{a^3 + a^2 + 2a}{a^3 - 1}$
$= \left[\dfrac{a + 1}{\left(a - 1\right) \left(a^2 + a + 1\right)} - \dfrac{1}{a^2 + a + 1} - \dfrac{2}{1 - a}\right] : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$
$= \left[\dfrac{a + 1 - a + 1}{\left(a - 1\right) \left(a^2 + a + 1\right)} + \dfrac{2}{a - 1}\right] : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$
$= \left[\dfrac{2}{\left(a - 1\right) \left(a^2 + a + 1\right)} + \dfrac{2}{a - 1}\right] : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$
$= \dfrac{2 + 2 a^2 + 2a + 2}{\left(a - 1\right) \left(a^2 + a + 1\right)} : \dfrac{a \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)}$
$= \dfrac{2a^2 + 2a + 4}{\left(a - 1\right) \left(a^2 + a + 1\right)} \times \dfrac{\left(a - 1\right) \left(a^2 + a + 1\right)}{a \left(a^2 + a + 2\right)}$
$= \dfrac{2 \left(a^2 + a + 2\right)}{\left(a - 1\right) \left(a^2 + a + 1\right)} \times \dfrac{\left(a - 1\right) \left(a^2 + a + 1\right)}{a \left(a^2 + a + 2\right)}$
$= \dfrac{2}{a}$ $\;\;\; \cdots \; (2)$
In view of $(2)$, expression $(1)$ becomes
$\dfrac{2}{a} - \dfrac{2}{a}$
$= 0$