I'm learning about series, and there this bit about partial fractions decomposition that I want to ask.
Am I right to assume that, there are multiple way to achieve the form $\frac{A}{1 - \alpha_1x}+\frac{B}{1 - \alpha_2x}$, i.e. there might be several solutions to $(1 - \alpha_1x)(1 - \alpha_2x) = 1 - 3x +x^2$, and hence to $A$ and $B$, but we choose the one most suited to simplify $H(x)$?
If this is true, is there some observation that can make choosing the right set of $\alpha_1,\alpha_2,A,B$ easier?
Your claim is not true. The partial fraction decomposition of any rational function is unique up to term order, as proven (for example) here, so there will always be only one possible solution for the parameters involved, again up to term order.