<s>[INST] Decompose $\frac{x^4 + 5}{x^5 + 6x^3}$ (partial fraction decomposition) 
Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients.
$$\frac{x^4 + 5}{x^5 + 6x^3}$$

So I factored the denominator to be $x^3(x^2+6)$ and here is the answer i got:
$\frac{Ax+B}{x^3}+\frac{Cx+D}{x^2+6}$
But this isn't the correct answer. Any help is appreciated!

Also have another one:

$\frac{5}{(x^2 − 16)^2}$

Here is what I got: $\frac{A}{x+4}+\frac{Bx+C}{\left(x+4\right)^2}+\frac{D}{x-4}+\frac{Ex+F}{\left(x-4\right)^2}$ which is also wrong... if anyone could help me that would be great
 [/INST] Question 1:
$$
x^{4}+5=f(x)\left(x^{2}+6\right)+x^{3} g(x)
$$
Let $ \quad  \displaystyle \frac{x^{4}+5}{x^{3}\left(x^{2}+6\right)} \equiv \frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x^{3}}+\frac{f(x)}{x^{2}+6}$,  then
$$x^{4}+5 \equiv A x^{2}\left(x^{2}+6\right)+B x\left(x^{2}+6\right)+C\left(x^{2}+6\right)+x^{3} f(x) \cdots (1)$$
Putting $x=0$ yields $5=6 C\Rightarrow C=\dfrac{5}{6}.$
Rearranging (1) yields
$$
\begin{aligned}
x^{4}+5-\frac{5}{6}\left(x^{2}+6\right) & \equiv A x^{2}\left(x^{2}+6\right)+B x\left(x^{2}+6\right)+x^{3} f(x) \\
\frac{1}{6} x\left(6 x^{2}-5\right) & \equiv A x\left(x^{2}+6\right)+B\left(x^{2}+6\right)+x^{2} f(x)\cdots (2)
\end{aligned}
$$
Putting $x=0$ in (2) yields $B=0$.
Rearranging (2) yields
$$
\frac{1}{6}\left(6 x^{2}-5\right)=A\left(x^{2}+6\right)+x f(x) \cdots (3)
$$
Putting $x=0$ in (3) yields $ \displaystyle -\frac{5}{6}=6 A \Rightarrow A=-\frac{5}{36}$.
$\begin{aligned}\text{ Then rearranging (3) yields } & \\ x^{2} f(x) &=\frac{1}{6}\left(6 x^{2}-5\right)+\frac{5}{36}\left(x^{2}+6\right) =\frac{41}{6} x^{2} \\ \therefore \quad f(x) &=\frac{41}{6} \\ \therefore   \frac{x^{4}+5}{x^{3}\left(x^{2}+6\right)} &=-\frac{5}{36 x}+\frac{5}{6 x^{3}}+\frac{41}{6\left(x^{2}+6\right)} .\end{aligned}$
Question 2:
For this particular fraction, we can do it in a simpler way. $$
\begin{aligned}
\frac{5}{\left(x^{2}-16\right)^{2}} &=5\left[\frac{1}{(x+4)(x-4)}\right]^{2} \\
&=5\left[\frac{1}{8}\left(\frac{1}{x-4}-\frac{1}{x+4}\right)\right]^{2} \\
&=\frac{5}{64}\left[\frac{1}{(x-4)^{2}}-\frac{2}{(x-4)(x+4)}+\frac{1}{(x+4)^{2}}\right] \\
&=\frac{5}{64}\left[\frac{1}{(x-4)^{2}}-\frac{1}{4}\left(\frac{1}{x-4}-\frac{1}{x+4}\right)+\frac{1}{(x+4)^{2}}\right] \\
&=\frac{5}{256}\left[\frac{4}{(x-4)^{2}}-\frac{1}{x-4}+\frac{1}{x+4}+\frac{4}{(x+4)^{2}}\right]
\end{aligned}
$$
</s>