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 $\frac{s-1}{s+1} \frac{s}{s^2+1} = \frac{A}{s+1} + \frac{Bs+C}{s^2+1}$ ?
$A = \left.\frac{s^2-s}{s^2+1} \right\vert_{s=-1} = \frac{1-(-1)}{1+1}=1$

Notice, $$\frac{s-1}{s+1}\frac{s}{s^2+1}=\frac{A}{s+1}+\frac{Bs+C}{s^2+1}$$
$$s^2-s=(A+B)s^2+(B+C)s+(A+C)$$
comparing the corresponding coefficients on both the sides, one should get $$A+B=1\tag 1$$
$$B+C=-1\tag 2$$
$$A+C=0\tag 3$$
on solving (1), (2) & (3), one can easily get  $\color{red}{A=1}, \color{blue}{B=0}, \color{red}{C=-1}$
