(a) To express \(u\) in the form \(x + iy\), multiply the numerator and denominator by the conjugate of the denominator \(1 - 2i\):

\(u = \frac{(\sqrt{2} - a\sqrt{2}i)(1 - 2i)}{(1 + 2i)(1 - 2i)}\)

The denominator becomes \(1^2 - (2i)^2 = 1 + 4 = 5\).

The numerator becomes \((1 - 2a)\sqrt{2} - (2 + a)\sqrt{2}i\).

Thus, \(u = \frac{1 - 2a}{5}\sqrt{2} + \frac{2 + a}{5}\sqrt{2}i\).

(b) With \(a = 3\), \(u = \frac{1 - 6}{5}\sqrt{2} + \frac{5}{5}\sqrt{2}i = -\frac{1}{5}\sqrt{2} + \sqrt{2}i\).

Calculate \(r = \sqrt{\left(-\frac{1}{5}\sqrt{2}\right)^2 + (\sqrt{2})^2} = 2\).

Calculate \(\theta = \arctan\left(\frac{\sqrt{2}}{-\frac{1}{5}\sqrt{2}}\right) = -\frac{3}{4}\pi\).

(c) The square roots of \(u\) are \(\sqrt{r}e^{i\theta/2}\) and \(\sqrt{r}e^{i(\theta/2 + \pi)}\).

\(\sqrt{2}e^{\frac{3}{8}\pi i}\) and \(\sqrt{2}e^{\frac{5}{8}\pi i}\).