5 In this question you must show detailed reasoning.
- Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
- Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
- Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.