Product expressions with roots

8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
Find the value of \(( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )\) in terms of \(k\).

4 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The roots of the equation $$x^3 - 8x^2 + 28x - 32 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the value of $$(\alpha + 2)(\beta + 2)(\gamma + 2).$$ [3 marks]