Find the coefficient of u^2 v^9 in the expansion of (2u^2 - v^3)^4.

Just expand it:

(2u^2 - v^3)^4 = (2u^2)^4 - 4(2u^2)^3 (v^3) + 6(2u^2)^2 (v^3)^2 - 4(2u^2) (v^3)^3 + (v^3)^4

The bold term is the term in u^2 v^9. You can just simplify it and you will see that the coefficient is -8.