Power series from points

Here's how you can find the power series of some unknown function through some points that you know the value of the function at. 

The main equation is: 

Say we're given the points:

Let's make the center point where we want the power series to be centered. This makes x=0, f(x)=2 because that's the y value at x=0, and f(x-2)=-6 from the leftmost point and so on. So:

after multiplying:

Now this can be solved for b, c, d. 

See what b,c,d represent is the value of the first, second, and third derivative at x=0 to substitute into the power series equation about x=0.