Explanation

(a)
Given, $x = \left( \frac{1 + i}{2} \right)$
$\Rightarrow \ \ 2x - 1 = i\ \ \Rightarrow \ 4x^{2} + 1 - 4x = - 1$
$\Rightarrow \ \ 2x^{2} - 2x + 1 = 0$
Since, $2x^{4} - 2x^{2} + x + 3$
$= \left( 2x^{2} - 2x + 1 \right)\left( x^{2} + x \right) + (3 - x^{2})$
$= 0 + 3 - \left( \frac{1 + i}{2} \right)^{2}$
$= 3 - \left( \frac{i}{2} \right)$