<s>[INST] How can this $T(n) = T(n-1)+T(n-2)+3n+1$ non homogenous recurrence relation be solved How are can the above recurrence relation be solved? 
I've reached here:
$(x^{2}-x-1)(x-3)^2(x-1)$
And then here:
$$a_n = l_1 \cdot (x_1)^n+l_2 \cdot (x_2)^n+l_3 \cdot (x_3)^n+l_4\cdot n \cdot (x_3)^n+l_5\cdot (x_4)^n$$
And we are given that these: $T(0) = 2, T(1) = 3$
So, for $n = 0$ and $n = 1$ we get two equations but we need 3 more, yet we don't have any more constants. 
 [/INST] Modify $T$ as follows:
$$
\begin{aligned}
T(n) &= T(n-1) + T(n-2) + 3n + 1\\
T(n) + 3n &= T(n-1) + T(n-2) + 6n + 1\\
T(n) + 3n &= T(n-1) + 3n + T(n-2) + 3n + 1\\
T(n) + 3n &= T(n-1) + 3(n-1) + T(n-2) + 3(n-2) + 10\\
T(n) + 3n + 10 &= T(n-1) + [3(n-1) + 10] + T(n-2) + [3(n-2) + 10 ]
\end{aligned}
$$
So, $S(n) = T(n) + 3n + 10$ satisfies the Fibonacci recurrence.
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