9.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-26_732_730_251_669}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

Figure 4 shows a sketch of the curve $C$ with equation $y = \mathrm { f } ( x )$, where

$$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$
\begin{enumerate}[label=(\alph*)]
\item Deduce the range of values of $x$ for which $\mathrm { f } ( x ) \geqslant 0$
\item Find $\mathrm { f } ^ { \prime } ( x )$ giving your answer in simplest form.

The point $R ( - 4,84 )$ lies on $C$.\\
Given that the tangent to $C$ at the point $P$ is parallel to the tangent to $C$ at the point $R$ (c) find the $x$ coordinate of $P$.\\
(d) Find the point to which $R$ is transformed when the curve with equation $y = \mathrm { f } ( x )$ is transformed to the curve with equation,
\begin{enumerate}[label=(\roman*)]
\item $y = \mathrm { f } ( x - 3 )$
\item $y = 4 \mathrm { f } ( x )$
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel PURE 2024 Q9}}