How to Solve Non-Homogeneous Recurrence Relations : $r_n = 2\left(r_{n-1} - \binom{n-1}{2}\right) + \binom{n-1}{2}$? $$r_n = 2\left(r_{n-1} - \binom{n-1}{2}\right) + \binom{n-1}{2}$$
which is equal to $$r_n - 2r_{n-1} = -\frac{n^2-3n+2}{2}$$
This given recurrence relation is derived from the question "How many regions line n could make at most in Euclidean plane?"
To solve this relation and make it into non-recurrent form, 
I had looked in Wikipedia but I only get to the point of solving the homogeneous part only.
$r_n - 2r_{n-1} = 0$ gives $\alpha \cdot2^n$ as a solution of homogenous part.
How should one deal with non-homogeneous part?

By induction easy to prove that
$$r_n=r_12^{n-1}+2^n-\frac{n^2+n+2}{2}.$$
Also, you can use the following telescopic sum, but it's very ugly, I think:
$$r_n=2r_{n-1}+\frac{(n-2)(n-1)}{2}$$
$$2^1r_{n-1}=2^2r_{n-2}+2^1\cdot\frac{(n-2)(n-3)}{2}$$
$$2^2r_{n-2}=2^3r_{n-3}+2^2\cdot\frac{(n-3)(n-4)}{2}$$
$$.$$
$$.$$
$$.$$
