<s>[INST] Partial Fractions Decomposition I am failing to understand partial fraction decomposition in cases like the following:
Provide the partial fraction decomposition of the following:
$$\frac{x+2}{(x-4)^3(x^2 + 4x + 16)}$$
I see this and I think of 
$$\frac{A}{x-4} + \frac{Bx+C}{(x-4)^2} + \frac{Dx^2 + Ex + F}{(x-4)^3} + \frac{Gx+H}{x^2 + 4x + 16}$$
But I am told that the correct answer is
$$\frac{A}{x-4} + \frac{B}{(x-4)^2} + \frac{C}{(x-4)^3} + \frac{Dx+E}{x^2 + 4x + 16}$$
What exactly is the numerator of each fraction based on?
 [/INST] Hint. Observe that you have
$$
\frac{Bx+C}{(x-4)^2} =\frac{B(x-4)+4B+C}{(x-4)^2}=\frac{B}{(x-4)}+\frac{4B+C}{(x-4)^2}
$$
and
$$
\begin{align}
\frac{Dx^2 + Ex + F}{(x-4)^3} &=\frac{D(x-4)^2+(E+8D)(x-4)+C+4E+32D}{(x-4)^3}\\\\
&=\frac{D}{(x-4)}+\frac{E+8D}{(x-4)^2}+\frac{C+4E+32D}{(x-4)^3}\\\\
\end{align}
$$
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