(i) Substitute \(z = -1 + i\) into \(p(z)\):

\(p(-1+i) = ((-1+i)^4) + 3((-1+i)^2) + 6(-1+i) + 10\).

Calculate \((-1+i)^2 = -1 + 2i - 1 = -2 + 2i\).

Calculate \((-1+i)^4 = ((-2+2i)^2) = 4 - 8i - 4 = -4 - 8i\).

Substitute back: \(p(-1+i) = (-4 - 8i) + 3(-2 + 2i) + 6(-1+i) + 10\).

\(= -4 - 8i - 6 + 6i - 6 + 6i + 10\).

\(= 0\).

Thus, \(u = -1 + i\) is a root.

(ii) Since \(-1 + i\) is a root, its conjugate \(-1 - i\) is also a root.

Find a quadratic factor with roots \(-1+i\) and \(-1-i\):

\((z + 1 - i)(z + 1 + i) = z^2 + 2z + 2\).

Divide \(p(z)\) by \(z^2 + 2z + 2\):

\(z^4 + 3z^2 + 6z + 10 = (z^2 + 2z + 2)(z^2 - 2z + 5)\).

Solve \(z^2 - 2z + 5 = 0\) using the quadratic formula:

\(z = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}\).

\(= \frac{2 \pm \sqrt{4 - 20}}{2}\).

\(= \frac{2 \pm \sqrt{-16}}{2}\).

\(= \frac{2 \pm 4i}{2}\).

\(= 1 \pm 2i\).

The other roots are \(-1 - i\), \(1 + 2i\), and \(1 - 2i\).

(i) The modulus of \(z\) is given by \(|z| = \sqrt{(\sqrt{2})^2 + (-\sqrt{6})^2} = \sqrt{2 + 6} = \sqrt{8} = 2\sqrt{2}\).

The argument of \(z\) is \(\arctan\left(\frac{-\sqrt{6}}{\sqrt{2}}\right) = \arctan(-\sqrt{3}) = -\frac{1}{3}\pi\).

(ii) (a) The complex conjugate \(z^* = \sqrt{2} + \sqrt{6}i\). Thus, \(z + 2z^* = (\sqrt{2} - \sqrt{6}i) + 2(\sqrt{2} + \sqrt{6}i) = 3\sqrt{2} + \sqrt{6}i\).

(b) \(\frac{z^*}{iz} = \frac{\sqrt{2} + \sqrt{6}i}{i(\sqrt{2} - \sqrt{6}i)} = \frac{\sqrt{2} + \sqrt{6}i}{i\sqrt{2} + 6}\).

Multiply numerator and denominator by the conjugate of the denominator: \(\frac{(\sqrt{2} + \sqrt{6}i)(-i\sqrt{2} + 6)}{(i\sqrt{2} + 6)(-i\sqrt{2} + 6)} = \frac{4\sqrt{3} + 4i}{8} = \frac{1}{2}\sqrt{3} + \frac{1}{2}i\).

(iii) On the Argand diagram, point \(A\) represents \(z^* = \sqrt{2} + \sqrt{6}i\) and point \(B\) represents \(iz = i(\sqrt{2} - \sqrt{6}i) = \sqrt{6} + i\sqrt{2}\).

The argument of \(\frac{z^*}{iz}\) is \(\frac{1}{2}\sqrt{3} + \frac{1}{2}i\), which corresponds to an angle of \(\frac{1}{6}\pi\).

1. Identify the points \(2i\) and \(-2+i\) on the Argand diagram.

2. The inequality \(|z - 2i| \leq |z + 2 - i|\) represents the region on or above the perpendicular bisector of the line segment joining \(2i\) and \(-2+i\).

3. The perpendicular bisector can be found by calculating the midpoint of \(2i\) and \(-2+i\), which is \(0 + \frac{1}{2}i\), and drawing a line perpendicular to the segment.

4. The inequality \(0 \leq \arg(z + 1) \leq \frac{1}{4}\pi\) represents the region within the angle formed by the positive real axis and the line with gradient 1 passing through \(-1, 0\).

5. Shade the region that satisfies both conditions: above the perpendicular bisector and within the specified angle.

(a) To solve the equation \((1 + 2i)w^2 + 4w - (1 - 2i) = 0\), we can use the quadratic formula \(w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1 + 2i\), \(b = 4\), and \(c = -(1 - 2i)\).

First, calculate the discriminant: \(b^2 - 4ac = 16 - 4(1 + 2i)(-1 + 2i)\).

Simplify \((1 + 2i)(-1 + 2i) = -1 + 2i - 2i - 4 = -5\).

Thus, \(b^2 - 4ac = 16 + 20 = 36\).

Now, apply the quadratic formula: \(w = \frac{-4 \pm \sqrt{36}}{2(1 + 2i)}\).

\(w = \frac{-4 \pm 6}{2 + 4i}\).

Calculate \(w = \frac{2}{2 + 4i}\) and \(w = \frac{-10}{2 + 4i}\).

Multiply numerator and denominator by the conjugate \(2 - 4i\):

\(w = \frac{2(2 - 4i)}{(2 + 4i)(2 - 4i)} = \frac{4 - 8i}{4 + 16} = \frac{4 - 8i}{20} = \frac{1}{5} - \frac{2}{5}i\).

\(w = \frac{-10(2 - 4i)}{20} = \frac{-20 + 40i}{20} = -1 + 2i\).

(b) On the Argand diagram, draw a circle centered at \(1 + i\) with radius \(2\). The region is bounded by the circle and the lines \(\arg z = -\frac{\pi}{4}\) and \(\arg z = \frac{\pi}{4}\), forming a sector.

(i) Obtain \(u + v = 1 + 2i\). In an Argand diagram, show points \(A, B, C\) representing \(u, v\) and \(u + v\) respectively. State that \(OB\) and \(AC\) are parallel and \(OB = AC\).

(ii) Multiply numerator and denominator of \(\frac{u}{v}\) by \(2 + i\) to simplify. Simplify the numerator to \(-5 + 5i\) and the denominator to 5. The final answer is \(-1 + i\).

(iii) Find \(\arg\left(\frac{u}{v}\right)\) to determine \(\angle AOB\). Show sufficient working to justify the given answer \(\frac{3}{4}\pi\).