a) EITHER: Substitute $1 + i\sqrt 3 $, attempt complete expansions of the ${x^3}$ and ${x^2}$ terms

Use ${i^2} =  - 1$ correctly at least once

Complete the verification correctly

State that the other root is $1 - i\sqrt 3 $

OR1: State that the other root is $1 - i\sqrt 3 $

State quadratic factor ${x^2} - 2x + 4$

Divide cubic by 3-term quadratic reaching partial quotient $2x + k$

Complete the division obtaining zero remainder

OR2: State factorisation $\left( {2x + 3} \right)\left( {{x^2} - 2x + 4} \right)$, or equivalent

Make reasonable solution attempt at a 3-term quadratic and use ${i^2} =  - 1$

Obtain the root $1 + i\sqrt 3 {\text{ }}$

State that the other root is $1 - i\sqrt 3 {\text{ }}$

b) Show point representing $1 + i\sqrt 3 $ in relatively correct position on an Argand diagram

Show circle with centre at $1 + i\sqrt 3 $ and radius 1

Show line for arg $z = \frac{1}{3}\pi $ making $\frac{1}{3}\pi $ with the real axis

Show line from origin passing through centre of circle, or the diameter which would contain the origin if produced

Shade the relevant region