2 The cubic equation

$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$

where $p , q$ and $r$ are real, has roots $\alpha , \beta$ and $\gamma$.\\
(a) Given that

$$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$

find the values of $p$ and $q$.\\
(b) Given further that one root is $3 + \mathrm { i }$, find the value of $r$.

\hfill \mbox{\textit{AQA FP2  Q2}}