Express $Q$ as a weighted combination of the five function values $f(a)$ through $f(e)$ and establish that its order is six. (See section 6.2.)
This is from Numerical Methods by Moler, http://www.mathworks.com/moler/quad.pdf Problem 6.2.
I'm having trouble understanding what exactly I need to do for it. My idea is to take basic simpsons and then take a division of it., would this essentially be $$S_2 = \frac h{12} (f(a) + 4f(d) + 2f(c) + 4f(e) + f(b)) \ ?$$ I pretty lost on establishing that the order is six too.
The exercise refers to the formula at the bottom of page 4: $$Q=S_2+(S_2-S)/15 \tag{1}$$ where $S$ is Simpson's rule (using three points $a,c,e$) and $S_2$ is composite Simpson's rule. Use the known formulas for $S$ and $S_2$ to obtain a formula for $Q$. (By the way, the method (1) is known as Richardson's extrapolation.)
The method the book uses to find the order of a quadrature method is to test the method on functions $x^0, x^1, x^2, \dots $. The first $k$ for which $x^k$ is not integrated exactly is the order of error of the method. You may find it convenient to do this computation on the interval $[-1,1]$ (which is related to any other interval by a linear change of variables): this speeds up the process considerably, because the odd powers of $x$ integrate to $0$, and it's easy to see that the quadrature method gives $0$ for them as well.