Problem

Figure 6 shows a sketch of part of the curve with equation $y=f(x)$, where

$$\begin{align*} f(x)=(2x+5)(x-3)^2. \end{align*}$$

Figure 6

(a) Deduce the values of $x$ for which $f(x)\leq0$.

The curve crosses the $y$-axis at the point $P$, as shown.

(b) Expand $f(x)$ to the form

$$\begin{align*} ax^3+bx^2+cx+d \end{align*}$$

where $a$, $b$, $c$ and $d$ are integers to be found.

(c) Hence, or otherwise, find

(i) the coordinates of $P$,

(ii) the gradient of the curve at $P$.

The curve with equation $y=f(x)$ is translated two units in the positive $x$ direction to a curve with equation $y=g(x)$.

(d) (i) Find $g(x)$, giving your answer in a simplified factorised form.

(ii) Hence state the $y$ intercept of the curve with equation $y=g(x)$.