How to prove that $f_{2k}+f_{2k-1}$ is a divisor of $xf_{4k}-2f_{4k-1}-2$

Question

Let $f_i$ are polynomials (of degree $i$) in variable $x$ which satisfies the following recurrence relation: $$f_{k+1}=xf_k-f_{k-1}\text{ for }k\ge1, \text{ and }$$ $$f_0=1, f_1=x.$$ How to prove that $f_{2k}+f_{2k-1}$ is a divisor of $xf_{4k}-2f_{4k-1}-2?$