The general quadratic curvature functional is considered in dimension four. It is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics. I'll show that a number of compact examples are infinitesimally rigid, and are therefore isolated as critical metrics. I'll also discuss solutions of the gauged linearized equation on several noncompact examples which are asymptotically locally Euclidean. This is joint work with Matt Gursky.