<s>[INST] Partial fraction of $\frac{s}{(s^2+2s+2)(s^2-2s+2)}$ I am trying to find the partial fraction of: 
$$\frac{s}{(s^2+2s+2)(s^2-2s+2)}$$
I started off with:
$$\frac{A(s^2-2s+2)}{s^2+2s+2} + \frac{B(s^2+2s+2)}{s^2-2s+2}$$
After that I get the following equations:
$A+B = 0$; $-2A+2B =1$
Giving: $A=-B$, and secondly $4B=1$, Hence:
$A=-\frac{1}{4}$ and $B=\frac{1}{4}$
But is this correct?
 [/INST] You want numerators of degrees lower than your denominators. Let's note that $$\frac{s}{(s^2+2s+2)(s^2-2s+2)}=\frac{1}{4}(\frac{1}{s^2-2s+2}-\frac{1}{s^2+2s+2}).$$ Complex numbers are required to obtain partial fractions with linear denominators, viz. $$\frac{1}{s^2\pm 2s+2}=\frac{1}{(s\pm 1)^2+1}=\frac{i}{2}(\frac{1}{s\pm 1 +i}-\frac{1}{s\pm 1 -i}).$$
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