10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{50ec901b-b6b6-4b72-85bd-a084f313c99b-22_592_665_251_676}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
Figure 6 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = ( 2 x + 5 ) ( x - 3 ) ^ { 2 }$$
- Deduce the values of \(x\) for which \(\mathrm { f } ( x ) \leqslant 0\)
The curve crosses the \(y\)-axis at the point \(P\), as shown.
- Expand \(\mathrm { f } ( x )\) to the form
$$a x ^ { 3 } + b x ^ { 2 } + c x + d$$
where \(a\), \(b\), \(c\) and \(d\) are integers to be found.
- Hence, or otherwise, find
- the coordinates of \(P\),
- the gradient of the curve at \(P\).
The curve with equation \(y = \mathrm { f } ( x )\) is translated two units in the positive \(x\) direction to a curve with equation \(y = \mathrm { g } ( x )\).
- Find \(\mathrm { g } ( x )\), giving your answer in a simplified factorised form.
- Hence state the \(y\) intercept of the curve with equation \(y = \mathrm { g } ( x )\).