Polynomial with parameter in coefficient

5. $$f ( x ) = x ^ { 3 } + ( p + 3 ) x ^ { 2 } - x + q$$ where \(p\) and \(q\) are constants and \(p > 0\) Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\)

1. show that $$9 p + q = - 51$$ Given also that when \(\mathrm { f } ( x )\) is divided by ( \(x + p\) ) the remainder is 9
2. show that $$3 p ^ { 2 } + p + q - 9 = 0$$
3. Hence find the value of \(p\) and the value of \(q\).
4. Hence find a quadratic expression \(\mathrm { g } ( x )\) such that $$f ( x ) = ( x - 3 ) g ( x )$$

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. $$f ( x ) = 4 x ^ { 3 } + 5 x ^ { 2 } - 10 x + 4 a \quad x \in \mathbb { R }$$ where \(a\) is a positive constant.
Given ( \(x - a\) ) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a \left( 4 a ^ { 2 } + 5 a - 6 \right) = 0$$
2. Hence
   1. find the value of \(a\)
   2. use algebra to find the exact solutions of the equation $$f ( x ) = 3$$