<s>[INST] How do you solve B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$? How do you solve  B and C for $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$ ?
$A = \left.\frac{s^2-s}{s^2+1} \right\vert_{s=-1} = \frac{1-(-1)}{1+1}=1$
 [/INST] We have
$$
\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1} \tag1
$$
Multiplying $(1)$ by $s$ and making $s \to \infty$ gives
$$
1=A+B
$$ from which $\color{red}{B=0}$.
Making $s=0$ in $(1)$ gives
$$
0=A+C
$$ from which $\color{red}{C=-1}$.
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