Verify that \(z ^ { 3 } - 8 = ( z - 2 ) \left( z ^ { 2 } + 2 z + 4 \right)\).
Solve the quadratic equation \(z ^ { 2 } + 2 z + 4 = 0\), giving your answers exactly in the form \(x + \mathrm { i } y\). Show clearly how you obtain your answers.
Show on an Argand diagram the roots of the cubic equation \(z ^ { 3 } - 8 = 0\).