Newton Divided Difference Problem

So my problem says:
Given the below data of polynomial P(x), determine the coefficient of all x² in P(x) if all third-order Newton divided difference are 1

x 0 1 2

P(x) 2 -1 4

What I did was find the First and second order divided differences. I think I might have done the problem incorrectly though, because I didn't use that the third order difference is 1. I was taught that the coefficient of x² would be the second order divided difference, so I just found that using the data points.

I got: f[x₀,x₁]=-3, f[x₁,x₂]=5, and f[x₀,x_1, x₂]=-4
I thought 4 would be the coefficient, but the back of the book says its 3.5. Any help?

Since the third order difference is constant, we know that we must have:



Since the third order difference is 1, we have:



Now, we are given:

and this results in the 2X2 system:





Solving this system, we find: