Q:

Did I do this big-O notation correctly?

It is for a homework assignment and I'm just getting thrown a little bit by the negative sign.

Express the following in terms of big-O notation. Use the tightest bounds possible. For instance, n5 is technically O(n1000), but this is not as tight as O(n5).
n2 −500n−2

n2 - 500 n - 2
<= n2 - 500 n
<= n2 for all n > 0
which is O(n2)

A:

For Big O notation what you need to remember is that it only matters for some number x0 and all numbers above that.  Specifically f(x)= O(g(x)) as x approaches infinity if there is some number M and some real number x0 such that |f(x)| <= M|g(x)| for all x >= x0. (Source for equations, wikipedia).
Basically, we only need to consider large values of x and you can pick an arbitrarily large value.  So large in fact that n^2 will overshadow a subtraction by 500n.  To be more technical if I pick M to be 2 and x0 to be 100000000000000000. Then the above equation holds.  I'm being lazy and picking an x0 that is extremely large but the equation lets me.  For an M equal to 2 a much smaller value of x0 would work, but again, it doesn't matter.
Finally, your answer of O(n^2) is correct

