Explanation

(a)
If $\omega$ and $\omega^{2}$ are two imaginary cube roots of unity.
Then, $1 + \omega + \omega^{2} = 0$
$\Rightarrow \ \omega + \omega^{2} = - 1$
Now, $a\omega^{317} + a\omega^{382} = a(\omega^{317} + \omega^{382})$
$= a\left( \omega^{2} + \omega \right) = - a$
And $a\omega^{317} \times a\omega^{382} = a^{2}\omega^{699} = a^{2}$
Therefore, the required equation is
$x^{2} - \left( a\omega^{317} + a\omega^{382} \right) + \left( a\omega^{317} \times a\omega^{382} \right) = 0$
$\Rightarrow x^{2} + ax + a^{2} = 0$