(i) Since the coefficients of the polynomial are real, the complex conjugate of any complex root must also be a root. Therefore, if -1 + \sqrt{3}i is a root, then -1 - \sqrt{3}i is also a root.

(ii) Substitute \(x = -1 + \sqrt{3}i\) into the equation \(kx^3 + 5x^2 + 10x + 4 = 0\) and expand \(x^2\) and \(x^3\) using \(i^2 = -1\).

Calculate \(x^2 = (-1 + \sqrt{3}i)^2 = 1 - 2\sqrt{3}i - 3 = -2 - 2\sqrt{3}i\).

Calculate \(x^3 = (-1 + \sqrt{3}i)(-2 - 2\sqrt{3}i) = 2 + 2\sqrt{3}i + 2\sqrt{3}i + 6i^2 = 2 + 4\sqrt{3}i - 6 = -4 + 4\sqrt{3}i\).

Substitute these into the equation:

\(k(-4 + 4\sqrt{3}i) + 5(-2 - 2\sqrt{3}i) + 10(-1 + \sqrt{3}i) + 4 = 0\).

Simplify and equate real and imaginary parts to zero to find \(k = 2\).

With \(k = 2\), the equation becomes \(2x^3 + 5x^2 + 10x + 4 = 0\).

\(Using the roots -1 + \sqrt{3}i and -1 - \sqrt{3}i, find the quadratic factor:\)

\((x + 1 - \sqrt{3}i)(x + 1 + \sqrt{3}i) = x^2 + 2x + 4\).

Divide the cubic by this quadratic to find the third root:

\(2x^3 + 5x^2 + 10x + 4 = (x^2 + 2x + 4)(2x + 1)\).

Thus, the third root is \(x = -\frac{1}{2}\).

(i) The complex number \(u = \sqrt{3} + i\) can be expressed in polar form as \(re^{i\theta}\). The modulus \(r\) is given by \(r = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{4} = 2\).

The argument \(\theta\) is \(\theta = \arctan\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}\).

Thus, \(u = 2e^{i\frac{\pi}{6}}\).

The modulus of \(u^4\) is \((2)^4 = 16\), and the argument of \(u^4\) is \(4 \times \frac{\pi}{6} = \frac{2\pi}{3}\).

(ii) Substitute \(u = \sqrt{3} + i\) into the equation \(z^3 - 8z + 8\sqrt{3} = 0\) and verify that it satisfies the equation. The other root is \(\sqrt{3} - i\).

(iii) On the Argand diagram, plot the point \(u = \sqrt{3} + i\). Draw a circle centered at \(u\) with radius 2. Draw the line \(y = 2\). Shade the region where \(|z - u| \leq 2\) and \(\text{Im } z \geq 2\).

(a) The given equation is \((1 + i)z^2 - (4 + 3i)z + 5 + i = 0\).

Use the quadratic formula: \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1+i\), \(b = -(4+3i)\), \(c = 5+i\).

Calculate \(b^2 = (4+3i)^2 = 16 + 24i + 9i^2 = 16 + 24i - 9 = 7 + 24i\).

Calculate \(4ac = 4(1+i)(5+i) = 4(5 + 6i - 1) = 16 + 24i\).

Then \(b^2 - 4ac = (7 + 24i) - (16 + 24i) = -9\).

\(\sqrt{-9} = 3i\).

Substitute into the quadratic formula:

\(z = \frac{4+3i \pm 3i}{2(1+i)}\).

\(z = \frac{4+6i}{2+2i}\) or \(z = \frac{4}{2+2i}\).

Multiply numerator and denominator by \(1-i\):

\(z = \frac{(4+6i)(1-i)}{2} = 1-i\).

\(z = \frac{4(1-i)}{2} = \frac{5}{2} + \frac{1}{2}i\).

(b) Plot the point \(u = -1 - i\) on the Argand diagram.

Draw the horizontal line through \(z = i\).

Draw half-lines from \(u\) with gradient 1 and vertical.

Shade the region satisfying \(|z| < |z - 2i|\) and \(\frac{1}{4}\pi < \text{arg}(z - u) < \frac{1}{2}\pi\).

(a)(i) Multiply the numerator and denominator by the conjugate of the denominator: \((1 - 2i)(1 + 2i) = 1 + 4 = 5\). The expression becomes \(\frac{(2 + 6i)(1 + 2i)}{5} = \frac{2 + 4i + 6i + 12i^2}{5} = \frac{2 + 10i - 12}{5} = \frac{-10 + 10i}{5} = -2 + 2i\).

(a)(ii) The modulus \(r\) is \(\sqrt{(-2)^2 + (2)^2} = \sqrt{8} = 2\sqrt{2}\). The argument \(\theta\) is \(\arctan\left(\frac{2}{-2}\right) = \arctan(-1) = \frac{3\pi}{4}\).

(b) The region is a circle centered at \(3i\) with radius 1. The greatest value of \(\arg z\) occurs at the intersection of the circle and the imaginary axis, which is at \(z = i\). The argument is \(\arg z = \frac{\pi}{2} + \sin^{-1}\left(\frac{1}{3}\right) \approx 1.91\).

(a) To express \(\frac{2 + 3i}{1 - 2i}\) in the form \(re^{i\theta}\), multiply numerator and denominator by the conjugate of the denominator:

\(\frac{2 + 3i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{(2 + 3i)(1 + 2i)}{(1 - 2i)(1 + 2i)}\)

Calculate the denominator: \((1 - 2i)(1 + 2i) = 1 + 2i - 2i - 4i^2 = 1 + 4 = 5\).

Calculate the numerator: \((2 + 3i)(1 + 2i) = 2 + 4i + 3i + 6i^2 = 2 + 7i - 6 = -4 + 7i\).

Thus, \(\frac{2 + 3i}{1 - 2i} = \frac{-4 + 7i}{5} = -\frac{4}{5} + \frac{7}{5}i\).

The modulus \(r\) is \(\sqrt{\left(-\frac{4}{5}\right)^2 + \left(\frac{7}{5}\right)^2} = \sqrt{\frac{16}{25} + \frac{49}{25}} = \sqrt{\frac{65}{25}} = \frac{\sqrt{65}}{5} \approx 1.61\).

The argument \(\theta\) is \(\arctan\left(\frac{7/5}{-4/5}\right) = \arctan\left(-\frac{7}{4}\right) \approx 2.09\).

(b) The equation \(|z - 3 + 2i| = 1\) represents a circle on the Argand diagram with center \((3, -2)\) and radius 1.

The least value of \(|z|\) is the distance from the origin to the closest point on the circle. The distance from the origin to the center \((3, -2)\) is \(\sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}\).

The least value of \(|z|\) is \(\sqrt{13} - 1\).