Just generalize your proof with .
This problem is again from I.M Gelfand's Algebra book.
Problem 157. Prove that for any polynomial of degree 2 all second differences are equal.
This proof isn't correct right? since isn't general case, if not, should I use induction to prove it?
Although this problem has been thoroughly solved, here is the outline of a different approach you might find interesting.
(1) Show that for a first degree polynomial all first differences are equal. (You may already have shown this, or it might be a theorem in your book.)
(2) Show that the difference operator applied to a second degree polynomial yields a first degree polynomial.
(1) and (2) combined show that the second differences of a second degree polynomial are equal.