(a) To find u and w, we start with the equations:

\(u - w = 2i\)

\(uw = 6\)

Substitute \(u = w + 2i\) into the second equation:

\((w + 2i)w = 6\)

\(w^2 + 2iw - 6 = 0\)

This is a quadratic equation in w. Solving using the quadratic formula:

\(w = \frac{-2i \pm \sqrt{(2i)^2 + 4 \times 6}}{2}\)

\(w = \frac{-2i \pm \sqrt{-4 + 24}}{2}\)

\(w = \frac{-2i \pm \sqrt{20}}{2}\)

\(w = \frac{-2i \pm 2\sqrt{5}}{2}\)

\(w = -i \pm \sqrt{5}\)

Thus, \(w = \sqrt{5} - i\) or \(w = -\sqrt{5} - i\).

Substituting back to find u:

If \(w = \sqrt{5} - i\), then \(u = w + 2i = \sqrt{5} + i\).

If \(w = -\sqrt{5} - i\), then \(u = w + 2i = -\sqrt{5} + i\).

(b) On the Argand diagram:

1. Plot the point \(2 + 2i\).

2. Draw a circle centered at \(2 + 2i\) with radius 2.

3. Draw a half-line from the origin at an angle of \(45^\circ\) to the positive x-axis.

4. Draw a vertical line at \(\text{Re } z = 3\).

5. Shade the region that satisfies all the inequalities.