In[3]:= RSolve[(a[n] == 3 a[n - 1] + 10 a[n - 2]), a[n], n]
Out[3]= {{a[n] -> (-2)^n C[1] + 5^n C[2]}}
Mathematica has (as far as I know) the best solver (available) for Recurrence Relations.
Without having done any work or thinking on my part: it seems you can use the answer to chooser your two Constants by the rule given. (or do more work to set constants, as the next person did)