6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 17 x - a$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 28 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\).

1. Find the values of \(a\) and \(b\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
3. State the number of roots of the equation \(\mathrm { p } \left( 2 ^ { y } \right) = 0\), justifying your answer.
   \includegraphics[max width=\textwidth, alt={}, center]{17025451-6f07-4f35-9dfc-869e084b5ed0-10_508_538_310_799} The diagram shows part of the curve $$y = 2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)$$ The shaded region is bounded by the curve and the two axes.
4. Show that \(2 \cos 2 x \cos \left( 2 x + \frac { 1 } { 6 } \pi \right)\) can be expressed in the form $$k _ { 1 } ( 1 + \cos 4 x ) + k _ { 2 } \sin 4 x ,$$ where the values of the constants \(k _ { 1 }\) and \(k _ { 2 }\) are to be determined.
5. Find the exact area of the shaded region.