VWO 1996 Problem 4

Consider a real poylnomial $p(x)=a_nx^n+...+a_1x+a_0$ . (a) If $\deg(p(x))>2$ prove that $\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x))$ . (b) Let $p(x)$ a polynomial for which there are real constants $r,s$ so that for all real $x$ we have $$p(x+1)+p(x-1)-rp(x)-s=0$$Prove $\deg(p(x))\le 2$ . (c) Show, in (b) that $s=0$ implies $a_2=0$ .