Hint: the formula will calculate the true value integrals of the form $\int_{-1}^{1} x^n |x| \,dx$ for all $n=0,1,...k$ such that $k$ is the accuracy degree.
How can I approach this?
An easy guess : $k=0$,$A_0 =1$ and $x_0 = 0$.
However, I don't know if the maximum value for $k$ is 0.
Each $n$ yield an equation $$ \int_{-1}^1 x^n |x| dx = A_0 x_0^n. $$ On the other hand, $$ \int_{-1}^1 x^n |x| dx = \begin{cases} \frac{2}{n+2}, &n \text{ is even}\\ 0, &n \text{ is odd} \end{cases} $$
So we have the following system of nonlinear equations for $A_0$ and $x_0$: $$ 1 = A_0, \quad n = 0\\ 0 = A_0 x_0, \quad n = 1\\ \vdots\\ \int_{-1}^1 x^k |x| dx = A_0 x_0^k, \quad n = k. $$ The first two equations immediately require $A_0 = 1, x_0 = 0$. If we truncate system at $n = 1$ then $A_0 = 1, x_0 = 0$ will be the solution. So $k$ is at least 1.
Let's check that $k$ cannot be greater than 1. If $k > 1$ then for $n = 2 \leq k$ the integration should be exact: $$ \frac{2}{4} = A_0 x_0^2. $$ But it is not true, so $k$ cannot be greater than $1$.