Explanation

(c)
$2\left( 1 + \frac{1}{\omega} \right)\left( 1 + \frac{1}{\omega^{2}} \right) + 3\left( 2 + \frac{1}{\omega} \right)\left( 2 + \frac{1}{\omega^{2}} \right) + \ldots + (n + 1)\left( n + \frac{1}{\omega} \right)\left( n + \frac{1}{\omega^{2}} \right)$
$= 2(1 + \omega)\left( 1 + \omega^{2} \right) + 3(2 + \omega)\left( 2 + \omega^{2} \right) + ... + (n + 1)(n + \omega)(n + \omega^{2})$
$= \sum_{r = 1}^{n}{(r + 1)(r + \omega)(r + \omega^{2})}$
$= \sum_{r = 1}^{n}{(r + 1)\lbrack r^{2} + \left( \omega + \omega^{2} \right)r + \omega^{3}\rbrack}$
$= \sum_{r = 1}^{n}{(r + 1)(r^{2} - r + 1)}$
$= \sum_{r = 1}^{n}{(r^{3} + 1)}$
$= \left\lbrack \frac{n(n + 1)}{2} \right\rbrack^{2} + n$