Solving the recurrence relation $a_k=3a_{k-1}+4a_{k-2}$ with $a_0=1$, $a_1=2$

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Solve the recurrence relation $a_k=3a_{k-1}+4a_{k-2}$ with $a_0=1$, $a_1=2$ using characteristic equation.

These are the steps I used. Is my work correct?

My work