Explanation

(d)
Given, $x = \frac{- 1 + \sqrt{3}i}{2} = \omega$
$\therefore\ \ \ \ \left( 1 - x^{2} + x \right)^{6} - \left( 1 - x + x^{2} \right)^{6}$
$= \left( 1 - \omega^{2} + \omega \right)^{6} - \left( 1 - \omega + \omega^{2} \right)^{6}$
$= \left( - 2\omega^{2} \right)^{6} - ( - 2\omega)^{6}\ \ \ \ \ \left\lbrack \because 1 + \omega + \omega^{2} = 0 \right\rbrack$
$= 2^{6}\omega^{12} - 2^{6}\omega^{6} = 0$ [$\because\omega^{3} = 1$]