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

\(u - w = 4i\)

\(uw = 5\)

Eliminate w by expressing it in terms of u: \(w = u - 4i\).

Substitute into the second equation: \(u(u - 4i) = 5\).

This gives the quadratic equation: \(u^2 - 4iu - 5 = 0\).

Solving this quadratic equation, we find:

\(u = 1 + 2i\) and \(u = -1 + 2i\).

Substituting back to find w:

If \(u = 1 + 2i\), then \(w = 1 - 2i\).

If \(u = -1 + 2i\), then \(w = -1 - 2i\).

Thus, the solutions are \(u = 1 + 2i, w = 1 - 2i\) or \(u = -1 + 2i, w = -1 - 2i\).

(b)(i) On the Argand diagram, plot the point \(2 - 2i\) and draw a circle centered at \(2 - 2i\) with radius 2.

Draw the line for \(\text{arg } z = -\frac{1}{4}\pi\) and the line \(\text{Re } z = 1\).

Shade the region satisfying all inequalities.

(b)(ii) The greatest possible value of \(\text{Re } z\) for points in the shaded region is \(2 + \sqrt{2}\).