4 It is given that \(\alpha , \beta\) and \(\gamma\) satisfy the equations
$$\begin{aligned}
& \alpha + \beta + \gamma = 1
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5
& \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 23
\end{aligned}$$
- Show that \(\alpha \beta + \beta \gamma + \gamma \alpha = 3\).
- Use the identity
$$( \alpha + \beta + \gamma ) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } - \alpha \beta - \beta \gamma - \gamma \alpha \right) = \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } - 3 \alpha \beta \gamma$$
to find the value of \(\alpha \beta \gamma\).
- Write down a cubic equation, with integer coefficients, whose roots are \(\alpha , \beta\) and \(\gamma\).
- Explain why this cubic equation has two non-real roots.
- Given that \(\alpha\) is real, find the values of \(\alpha , \beta\) and \(\gamma\).