9. (a) Given that $$\frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \equiv x ^ { 2 } + P + \frac { Q } { x - 4 } \quad x > - 3$$ find the value of the constant \(P\) and show that \(Q = 5\) The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$g ( x ) = \frac { x ^ { 4 } - x ^ { 3 } - 10 x ^ { 2 } + 3 x - 9 } { x ^ { 2 } - x - 12 } \quad - 3 < x < 3.5 \quad x \in \mathbb { R }$$ (b) Find the equation of the tangent to \(C\) at the point where \(x = 2\) Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = 2\)
(c) Find the exact area of \(R\), writing your answer in the form \(a + b \ln 2\), where \(a\) and \(b\) are constants to be found. \includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-31_2255_50_314_34}
\includegraphics[max width=\textwidth, alt={}, center]{96948fd3-5438-4e95-b41b-2f649ca8dfac-32_106_113_2524_1832}