$\displaystyle s = \frac{s^2+8}{(s^2+4)(s^2+9)}$
$\displaystyle A(s^2+9) + B(s^2+4) = s^2+8$
Uncertain of what number to substitute to solve ?
You had $\displaystyle A(s^2+9) + B(s^2+4) = s^2+8$. Multiply it out: $\displaystyle As^2+ 9A+ Bs^2+ 4B= s^2+ 8$
$\displaystyle (A+ B)s^2+ (9A+ 4B)= s^2+ 8$
The coefficient of $\displaystyle s^2$ is A+ B on one side, 1 on the other: you must have A+ B= 1. The coefficient of s is 0 on both sides (whew!). The constant term is 9A+ 4B on one side and 8 on the other. You must have 9A+ 4B= 8.