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**awkward** 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.