<s>[INST] Decomposition of $\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}$ 
Determine $\alpha,\beta,\gamma,\delta$ in
$$\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{\alpha s+\beta}{(s+3)^2+25}+\frac{\gamma s+\delta}{s^2+25}$$

I have come across the partial fraction and trying to decompose it. I have tried the methods used by multiplying out and making one variable to 0 to find the others etc but It does not work here. Any ideas how I could approach this?
 [/INST] Apply your method to proceed as follows
$$\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}
=\frac{A}{s+5i} + \frac{B}{s-5i} +\frac{C}{s+3+i5}+ \frac{D}{s+3-i5}$$
Then
\begin{align}
&A = \lim_{s\to-5i} \frac{10(s+6)(s+5i)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{-45+68i}{327}\\
&B = \lim_{s\to5i} \frac{10(s+6)(s-5i)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{-45-68i}{327}\\
&C = \lim_{s\to-3-5i} \frac{10(s+6)(s+3+5i)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{45-41i}{327}\\
&D= \lim_{s\to-3+5i} \frac{10(s+6)(s+3-5i)}{\left[(s+3)^2+25\right](s^2+25)}=\frac{45+41i}{327}\\
\end{align}
and
\begin{align}
& \lambda = A+B = -\frac{90}{327}\\
& \delta =(B -A)\>5i= \frac{680}{327}\\
 & \alpha = C +D = \frac{90}{327}\\
& \beta =(C+D) \>3+(D -C ) \>5i= -\frac{140}{327}\\
\end{align}
Thus
$$\frac{10(s+6)}{\left[(s+3)^2+25\right](s^2+25)}
=\frac{10}{327}\left( \frac{68-9s}{s^2+25} + \frac{9s-14}{(s+3)^3+25}\right)$$
</s>