The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot 
I understand how to take the Taylor series in the hint, but the 4 expressions in the numerator of the rule are throwing me off. Would I separate the fraction into two fractions? And then add the error terms? That doesn't seem right to me...
Or would there be 4 different coefficients for each term in the expansion? Again, that doesn't seem right to me...
Consider, around $h=0$, Taylor expansion $$f(a+\alpha h)=f(a)+\alpha h f'(a)+\frac{1}{2} \alpha ^2 h^2 f''(a)+\frac{1}{6} \alpha ^3 h^3 f^{(3)}(a)+\frac{1}{24} \alpha ^4 h^4 f^{(4)}(a)+\frac{1}{120} \alpha ^5 h^5 f^{(5)}(a)+O\left(h^6\right)$$ Apply it to $\alpha=+2,+1,-1,-2$ to get $$f(a+2 h)-2 f(a+ h)+2 f(a- h)-f(a-2 h)=2 h^3 f^{(3)}(a)+\frac{1}{2} h^5 f^{(5)}(a)+O\left(h^6\right)$$
Using the hint, calculations can be faster since $$f(a+\alpha h)-f(a-\alpha h)=2 \alpha h f'(a)+\frac{1}{3} \alpha ^3 h^3 f^{(3)}(a)+\frac{1}{60} \alpha ^5 h^5 f^{(5)}(a)+O\left(h^6\right)$$