Question 9 - A-Level Maths

Show that \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 } \equiv \alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + 2 \alpha \beta \gamma ( \alpha + \beta + \gamma )\).
It is given that \(\alpha , \beta\) and \(\gamma\) are the roots of the cubic equation \(x ^ { 3 } + p x ^ { 2 } - 4 x + 3 = 0\), where \(p\) is a constant. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\) in terms of \(p\).