The complex numbers \(u\) and \(w\) satisfy the equations
$$u - w = 4 \mathrm { i } \quad \text { and } \quad u w = 5$$
Solve the equations for \(u\) and \(w\), giving all answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
On a sketch of an Argand diagram, shade the region whose points represent complex numbers satisfying the inequalities \(| z - 2 + 2 \mathrm { i } | \leqslant 2 , \arg z \leqslant - \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \geqslant 1\), where \(\operatorname { Re } z\) denotes the real part of \(z\).
Calculate the greatest possible value of \(\operatorname { Re } z\) for points lying in the shaded region.