Partial fraction of a generating function

Question

I am solving a recurrence relation

$a_0 = a_1 = a_2 = 1, a_{n+3} = a_{n+2} − 2a_{n+1} − 4a_n$ for $n \ge 0$

I got a generating function for this sequence

$f(x) = \frac{2x^2+1}{4x^3+2x^2-x+1} $

Now I want to get the formula for the n-th term of this sequence. And so I wanted to use the partial fractions method but so far I was unsuccesful in finding the partial fraction. I'm probably missing something simple here.

Answer 1

hint:

$$4x^3 +2x^2 -x +1=(4x^2 -2x +1) (x+1) $$