\begin{enumerate} \item (a) Using the factor theorem, show that \(( x + 3 )\) is a factor of \(x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24\).
(b) Factorise \(x ^ { 3 } - 3 x ^ { 2 } - 10 x + 24\) completely. \item (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
(b) Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers. \item The first three terms in the expansion, in ascending powers of \(x\), of \(( 1 + p x ) ^ { n }\), are \(1 - 18 x + 36 p ^ { 2 } x ^ { 2 }\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\).
[0pt] [P2 January 2003 Question 2] \item A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 8 y - 75 = 0\).