Impulse response of a second order LTI
Your procedure is correct. But, you need to consider two things:

(i) Pole/zero locations.

(ii) Whether the system is causal or non-causal.

Your final answer is OK provided you consider that all the zeros and poles are distinct and the underlying LTI system is causal.

You will get two more solutions if the LTI system is considered non-causal, even if you assume to have distinct zeros and poles.

Hope this helps.