Explanation

(a)
Since, $\alpha\ and\ \beta$ are the roots of the equation $x^{2} + px + q = 0$, therefore
$\alpha + \beta = - p\ and\ \alpha\beta = q$
Now, $\left( \omega\alpha + \omega^{2}\beta \right)(\omega^{2}\alpha + \omega\beta)$
$= \alpha^{2} + \beta^{2} + (\omega^{4} + \omega^{2})\alpha\beta$ $(\because\omega^{3} = 1)$
$= \alpha^{2} + \beta^{2} - \alpha\beta$ $(\because\ \omega + \omega^{2} = - 1)$
$= (\alpha + \beta)^{2} - 3\alpha\beta$
$= p^{2} - 3q$
Also, $\frac{\alpha^{2}}{\beta} + \frac{\beta^{2}}{\alpha} = \frac{\alpha^{3} + \beta^{3}}{\alpha\beta}$
$= \frac{(\alpha + \beta)^{3} - 3\alpha\beta(\alpha + \beta)}{\alpha\beta}$
$= \frac{p(3q - p^{2})}{q}$
$\therefore\$ The given expression $= \frac{(p^{2} - 3q)}{\frac{p(3q - p^{2})}{q}} = - \frac{q}{p}$