First, you need to make more sense. How can a polynomial have a common difference? What does that mean?
On the other hand, your answer may be hiding in successive derivatives.
If p(x) = ax^5 + 3x^4 - 2x^3 - 3x^2 + x - 1
Then p'(x) = 5ax^4 + 12x^3 - 6x^2 - 6x^1 + 1
Then p''(x) = 20ax^3 + 36x^2 - 12x^1 - 6
Then p'''(x) = 60ax^2 + 72x^1 - 12
Then p''''(x) = 120ax^1 + 72
Then p'''''(x) = 120a = 120