<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] Cross-multiplying on the right hand side gives
$As^{2} + A + Bs^{2} + Bs + Cs + C$ on the numerator. 
 Plug A = 1
$(1 + B)s^{2} + (B + C)s + (1 + C) = s^{2} - s$.  Equate the coefficients
$1 + B = 1 \rightarrow B = 0$
$B + C = -1 \rightarrow C = -1$
</s>