If I understand your question you have

$f(x)=x^2-x-2$

and you want to find the Riemann Sum approximation of

$\displaystyle{\int_{-2}^{2}}f(x)~dx$

With $n=4$ It should be pretty clear that your left endpoints will be $-2, -1, 0, 1$, and that $\Delta x=1$

This matches with what you found.

So your Riemann sum is

$f(-2)+f(-1)+f(0)+f(1) = 4+0-2-2=0$

how you got 60 out of this I don't know.

The actual definite integral above is equal to $-\dfrac 8 3$.

The Riemann Sum is just an approximation. If you let n=8 you find the Riemann Sum equals $-\dfrac 3 2$, i.e.

the approximation improves with increasing $n$ as is expected.