I am struggling to find a means of tackling question (a). Initially I tried to find the original piecewise pdf but this appears to be over complicating the problem. Would it make sense to subtract the $cx^2$ from $1 - c(4-x)^2$ and find the derivative of this and find the derivative of $cx^2$ in order to get the pdf and then integrate over the bounds in order to find $c$?
First of all, nothing in the question indicates that the cdf $F(x)$ is continuous, so any $0\leq c\leq \frac18$ is valid. For example, for $c=0$ we get the valid cdf $$ F(x)=\begin{cases}0, & x< 2\cr 1, & x\geq 2\end{cases} $$
If you need to obtain continuous cdf, take left limit at $x=2$ and equate it to the value at $x=2$. You will get $c=\frac18$.