Polynomial functions - finite difference - leading coefficient

Can someone explain to me how you can find out what the leading coefficient is from a finite difference.

For example, the function

ax^5 + 3x^4 - 2x^3 - 3x^2 + x - 1

has a common difference of -120.

What is the value of a?

You don't have to tell me the answer of this question, just the process of how to find it. I want to be able to figure this out myself, but I have no clue where to start. Any hints or leads are GREATLY APPRECIATED!!!

Thanks!