10 The equation
$$x ^ { 4 } + x ^ { 3 } + c x ^ { 2 } + 4 x - 2 = 0$$
where \(c\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\).
- Use the substitution \(y = \frac { 1 } { x }\) to find an equation which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }\).
- Find, in terms of \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\) and \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
- Hence find
$$\left( \alpha - \frac { 1 } { \alpha } \right) ^ { 2 } + \left( \beta - \frac { 1 } { \beta } \right) ^ { 2 } + \left( \gamma - \frac { 1 } { \gamma } \right) ^ { 2 } + \left( \delta - \frac { 1 } { \delta } \right) ^ { 2 }$$
in terms of \(c\).
- Deduce that when \(c = - 3\) the roots of the given equation are not all real.