4 The polynomial \(\mathrm { p } ( x )\) is defined by
$$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$
where \(a\) is a constant.
- Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) for all values of \(a\).
- Given that the remainder is - 42 when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ), find the value of \(a\).
- When \(a\) has the value found in part (ii), factorise \(\mathrm { p } \left( x ^ { 2 } \right)\) completely.