7 The diagram shows the sketch of a curve and the tangent to the curve at \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{0d5b9235-af2b-4fd5-8fcf-b2b45e3c0a3c-14_519_817_356_614}
The curve has equation \(y = 4 - x ^ { 2 } - 3 x ^ { 3 }\) and the point \(P ( - 2,24 )\) lies on the curve. The tangent at \(P\) crosses the \(x\)-axis at \(Q\).
- Find the equation of the tangent to the curve at the point \(P\), giving your answer in the form \(y = m x + c\).
- Hence find the \(x\)-coordinate of \(Q\).
- Find \(\int _ { - 2 } ^ { 1 } \left( 4 - x ^ { 2 } - 3 x ^ { 3 } \right) \mathrm { d } x\).
- The point \(R ( 1,0 )\) lies on the curve. Calculate the area of the shaded region bounded by the curve and the lines \(P Q\) and \(Q R\).
[0pt]
[3 marks]