I need to find A,B,C s.t
(1) $\int_{x_0-2h}^{x_0} f(x) dx=Af(x_0-2h)+Bf(x_0)+C(f'(x_0)-f'(x_0-2h))+O(h^4)$
My first guess was using Simpson's rule, but I can't use $f(x_0-h)$.
Using Taylor I managed to evaluate $f(x_0-h)$ by
$f(x_0-h)=f(x_0)-f'(x_0)h + \frac{f''(\xi_1)h^2}{2}$ $f(x_0-h)=f(x_0-2h) + f'(x_0-2h)h + \frac{f''(\xi_2)h^2}{2}$
(2) $f'(x_0)-f'(x_0-2h) = f(x_0)+f(x_0-2h)-2f(x_0-h)+O(h^2)$
then I replaced $f'(x_0)-f'(x_0-2h)$ in (1) and tried to find A, B and C that match Simpson's rule. I got an error of $O(h^3)$