The polynomial $4{x^3} + \alpha {x^2} + 9x + 9$, where $\alpha $ is a constant, is denoted by $p\left( x \right)$. It is given that when $p\left( x \right)$ is divided by $\left( {2x - 1} \right)$ the remainder is $10$.
a) Find the value of $\alpha $ and hence verify that $\left( {x - 3} \right)$ is a factor of $p\left( x \right)$.
b) When $\alpha $ has this value, solve the equation $p\left( x \right) = 0$.
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