a) Attempt to solve for $m$ the equation $p\left( { - 2} \right) = 0$ or equivalent

Obtain $m = 6$

Alternative:

Attempt $p\left( z \right) \div \left( {z + 2} \right)$, equate a constant remainder to zero and solve for $m$.

Obtain $m = 6$

b)(i) State $z =  - 2$

Attempt to find quadratic factor by inspection, division, identity, …

Obtain ${z^2} + 4z + 16$

Use correct method to solve a 3-term quadratic equation

Obtain $ - 2 \pm 2\sqrt {3i} $ or equivalent

(ii) State or imply that square roots of answers from part (b)(i) needed

Obtain $ \pm i\sqrt 2 $

Attempt to find square root of a further root in the form $x + iy$ or in polar form

Obtain ${\alpha ^2} - {b^2} =  - 2$ and $\alpha b = \left(  \pm  \right)\sqrt 3 $ following their answer to part (b)(i)

Solve for $\alpha $ and $b$

Obtain $ \pm \left( {1 + i\sqrt 3 } \right)$ and $ \pm \left( {1 - i\sqrt 3 } \right)$