Help solving a non-homogenous recurrence relation

Question

I'm given the following recurrence relation: $a_n = 4a_{n-1} - 4a_{n-2} + (n+1)2^n$. I've solved it using generating functions, and am supposed to use characteristic equations to solve it. I know that the characteristic root is 2, so my "guessed" solution must be $n^2(An+B)2^n$. I've plugged this into my relation, but have no idea how to proceed from here. As far as I can tell, I'll be left with a single $An^3$ term, along with a constant term. Therefore, I can't solve it by equating coefficients, and nor can I solve it by setting the equation to 0. Am I missing something? I've looked at other questions on here, and I can't figure out how to do the transformations some of them suggest. I'd prefer a solution that is more in line with the method I tried to use. Thanks!

Answer 1

Hint: $\;\; \require{cancel} \dfrac{a_{n+1}}{2^{n+1}} - \dfrac{a_n}{2^n} = \left(2\dfrac{a_{n}}{2^n} - \dfrac{a_{n-1}}{2^{n-1}} + \cancel{n}+2\right) - \left(2\dfrac{a_{n-1}}{2^{n-1}} - \dfrac{a_{n-2}}{2^{n-2}} + \cancel{n} +1\right) \;\;$ is a linear recurrence for $\,b_n = \dfrac{a_n}{2^n}\,$.