To find the impulse repsonse using the difference equation
There is no real shortcut for computing the impulse response. Partial fractions is the standard way to do it. However, in the case of a second-order transfer function as in your example, the result can be found in tables of $\mathcal{Z}$-transform pairs. E.g., if you use the last two correspondence of this table, you can pretty quickly write down the result.

By the way, your result can't be correct (at least not for negative $n$), because you know that the system is causal, so its impulse response must be zero for $n&lt;0$.