<s>[INST] Expanding $1/[(s^2+1)(s+1)]$ into partial fractions by brute force I want to do partial fractions on: $1/[(s^2+1)(s+1)]$
$(as+b)/(s^2+1) + c/(s+1)$
Multiplying both sides by $(s^2+1)(s+1)$, get $a=-1/2, b=1/2, c=1/2$
$(-s/2+1/2)/(s^2+1) + (1/2)/(s+1) = 1/[(s^2+1)(s+1)]$
Unfortunately when I add $(-s/2+1/2)/(s^2+1) + (1/2)/(s+1)$ I get $(1/4)/[(s^2+1)(s+1)]$.
Where did I go wrong?
 [/INST] No,when you add 
$$
\frac{-s/2+1/2}{s^2+1} + \frac{1/2}{s+1}
$$
you get
$$
\frac{1}{2}\frac{1-s^2+1}{(s^2+1)(s+1)}
$$
or 
$$
\frac{1}{(s+1)(s^2+1)}.
$$
</s>