If you refer to my picture:
https://i.stack.imgur.com/lVsU1.png
I'm having a hard time understanding why in the 2nd step the fraction is split up in two terms when 2 is a constant. I get why you replace the s in the numerator with (s-1), but i'm not sure how 2 becomes its own fraction. When first solving this problem I thought you could just pull the 2 out because its a constant but thats wrong.
So you have $$\frac {2s}{(s-1)^2 + 4}$$ and you want to express everything in terms of $s-1$.
It might help to call that by a different variable name, say $r:= s -1$. So $s = r+1$ and the expression becomes $$\begin{align}\frac {2(r+1)}{r^2 + 4}&= \frac {2r}{r^2 + 4} + \frac 2{r^2 + 4}\\&=\frac {2(s-1)}{(s-1)^2 + 4} + \frac 2{(s-1)^2 + 4}\end{align}$$