The polynomial ${x^3} + 4{x^2} + \alpha x + 2$, where $\alpha $ is a constant, is denoted by $p\left( x \right)$. It is given that the remainder when $p\left( x \right)$ is divided by $\left( {x + 1} \right)$ is equal to the remainder when $p\left( x \right)$ is divided by $\left( {x - 2} \right)$.
a) Find the value of $\alpha $.
b) When $\alpha $ has this value, show that $\left( {x - 1} \right)$ is a factor of $p\left( x \right)$ and find the quotient when $p\left( x \right)$ is divided by $\left( {x - 1} \right)$.
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