8. The curve \(C\) has equation \(y = 4 x + 3 x ^ { \frac { 3 } { 2 } } - 2 x ^ { 2 } , \quad x > 0\).
- Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Show that the point \(P ( 4,8 )\) lies on \(C\).
- Show that an equation of the normal to \(C\) at the point \(P\) is
$$3 y = x + 20 .$$
The normal to \(C\) at \(P\) cuts the \(x\)-axis at the point \(Q\).
- Find the length \(P Q\), giving your answer in a simplified surd form.