Answer is done,
The answer is done, but I just need a detailed step-by-step how to get there. I'm still confused. Please show me step-by-step.

Answer
First note that $f$ has to be continuous. Hence we should have
\[\lim_{x\rightarrow 4^-}f(x)=\lim_{x\rightarrow 4^+}f(x) \Rightarrow c(4)^2=10-4d.\]
Thus
\[16c=10-4d. (1)\]
For $x< 4$, we have
\[f'(x)=2cx \Rightarrow \lim_{x\rightarrow 4^-}f'(x)=2c(4)=8c.\]
For $x>4$ we have
\[f'(x)=-d \Rightarrow \lim_{x\rightarrow 4^+}f'(x)=-d.\]
For $f'$ to be continuous, we should have
\[\lim_{x\rightarrow 4^-}f'(x)=\lim_{x\rightarrow 4^+}f'(x) \Rightarrow 8c=-d.\]
Thus
\[8c=-d. (2)\]
Solving the symstem of equations (1) and (2), we have
\[16c=10-4d, 16c=-2d \Rightarrow 10-4d=-2d\]
\[10=2d \Rightarrow d=5. \]
Also
\[16c=-2(5)=-10. \Rightarrow c=-\frac{-10}{16}=-\frac{5}{8}.\]

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