(a)[6]
Show every step and, without a calculator, solve the equation $(1 + i)z^2 - (4 + 3i)z + 5 + i = 0$. Present your solutions in the form $x + iy$, where $x$ and $y$ are real.
(b)[4]
The complex number $u$ is $u = -1 - i$. On a sketch of an Argand diagram, mark the point for $u$. Shade the region whose points represent complex numbers satisfying the inequalities $|z| < |z - 2i|$ and $\frac{1}{4}\pi < \arg(z - u) < \frac{1}{2}\pi$.