Is the derivative of
Or did I screw up?
Sadly, I think you screwed up. When you brought the second power down, remember that you're just reducing the power by one, i.e., the parenthetical is still being raised to the first power.
Note your original equation is a fourth-degree polynomial (since it has an x-squared term squared), so your derivative should be a third degree polynomial.
You can multiply the parts of the derivative in whatever order you want - multiplication is commutative and can be performed in any order. I chose to FOIL multiply the two binomials together (combining the outer and inner terms, as they were both x^2 terms) and then lastly distributing the 6 through.