4 The roots of the equation
$$z ^ { 3 } + 2 z ^ { 2 } + 3 z - 4 = 0$$
are \(\alpha , \beta\) and \(\gamma\).
- Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
- Hence show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 2\).
- Find the value of:
- \(( \alpha + \beta ) ( \beta + \gamma ) + ( \beta + \gamma ) ( \gamma + \alpha ) + ( \gamma + \alpha ) ( \alpha + \beta )\);
- \(( \alpha + \beta ) ( \beta + \gamma ) ( \gamma + \alpha )\).
- Find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).