Sum of powers of roots

The roots of the equation \(x^4 - 2x^3 + 2x^2 + x - 3 = 0\) are \(\alpha\), \(\beta\), \(\gamma\) and \(\delta\). Determine the values of

1. \(\alpha^2 + \beta^2 + \gamma^2 + \delta^2\), [2]
2. \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta}\), [2]
3. \(\alpha^3 + \beta^3 + \gamma^3 + \delta^3\). [4]

The equation $$8x^3 + 12x^2 + 4x - 1 = 0$$ has roots \(\alpha, \beta, \gamma\). Show that the equation with roots \(2\alpha + 1, 2\beta + 1, 2\gamma + 1\) is $$y^3 - y - 1 = 0.$$ [3] The sum \((2\alpha + 1)^n + (2\beta + 1)^n + (2\gamma + 1)^n\) is denoted by \(S_n\). Find the values of \(S_3\) and \(S_{-2}\). [5]