Explanation

(a)
The given equation is
$2(1 + i)x^{2} - 4(2 - i)x - 5 - 3i = 0$
$\Rightarrow \ x = \frac{4(2 - i) \pm \sqrt{16(2 - i)^{2} + 8(1 + i)(5 + 3i)}}{4(1 + i)}$
$= \frac{i}{1 + i}$ or $\ \frac{4 - i}{1 + i}$
$= \frac{- 1 - i}{2}\ or\ \frac{3 - 5i}{2}$
Now, $\left| \frac{- 1 - i}{2} \right| = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{1}{2}}$
and $\left| \frac{3 - 5i}{2} \right| = \sqrt{\frac{9}{4} + \frac{25}{4}} = \sqrt{\frac{17}{2}}$
Also, $\sqrt{\frac{17}{2}} > \sqrt{\frac{1}{2}}$
Hence, required root is $\frac{3 - 5i}{2}.\$