Finding probabilities from probabilty generating function

Question

Given that I have a probability generating function for $Q$ given by $\dfrac{4s^{2}}{9-3s-2s^{2}}$, I want to find $P(Q = n)$ for $n \geq 2$. I understand that I could actually use the definition of the probability generating function and derivatives, but that seems crazy. Is there a way to turn this into a summation so I can take the coefficient possibly?

Answer 1

Since $2s^2+3s-9=(s+3)(2s-3)$, we have:

$$\frac{4s^2}{9-3s-2s^2}=-2+\frac{4}{s+3}-\frac{1}{s-3/2}\tag{1}$$ and now we just need to exploit: $$\forall \alpha:|\alpha|<1,\qquad \frac{1}{1-\alpha s} = \sum_{n\geq 0} \alpha^n s^n. \tag{2}$$