I know that in general, polynomial satisfiability is NP; however, I'm curious to know what work has been done on special classes of polynomials, and in particular quadratic polynomials of multiple variables.

EDIT: Just to clarify, I want to know if the polynomial has a solution over the real numbers. By satisfiability, I mean if a system of polynomials has a solution, but here I am only concerned with one quadratic multivariate polynomial having a solution over the reals.