NL-NWO 1963 Problem 4

One considers for $n > 2$ the polynomial: $$(x^2-x+1)^n - (x^2-x+2)^n+ (1+x)^n+(2-x)^n$$ Show that the degree of this polynomial is $2n - 2$ . The polynomial is written in the form $$a_0+a_1x+a_2x^2+...+a_{2n-2}x^{2n-2}$$ Prove that $a_2+a_3+...+a_{2n-2}=0$