Show that \(( 2 + \mathrm { i } ) ^ { 3 }\) can be expressed in the form \(2 + b \mathrm { i }\), where \(b\) is an integer.
It is given that \(2 + \mathrm { i }\) is a root of the equation
$$z ^ { 3 } + p z + q = 0$$
where \(p\) and \(q\) are real numbers.
Show that \(p = - 11\) and find the value of \(q\).
Given that \(2 - \mathrm { i }\) is also a root of \(z ^ { 3 } + p z + q = 0\), find a quadratic factor of \(z ^ { 3 } + p z + q\) with real coefficients.
Find the real root of the equation \(z ^ { 3 } + p z + q = 0\).