8 The cubic polynomial \(2 x ^ { 3 } + k x ^ { 2 } - x + 6\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
- Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
- Find \(\int _ { - 1 } ^ { 2 } f ( x ) \mathrm { d } x\).
- Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 1 \leqslant x \leqslant 2\).
\section*{[Question 9 is printed overleaf.]}