Let $(f_n)$ be the Fibonacci sequence: $f_1=f_2=1$, $f_{n}=f_{n-1}+f_{n-2}$ for all $n>2$. Prove that for every odd $n\geq 3$ the polynomial $\ \ x^{n}+f_{n}x^2-f_{n-2}\ \ $ is divisible by $x^2+x-1$.