  We study two large classes of alternative theories, modifying the action
through algebraic, quadratic curvature invariants coupled to scalar fields. We
find one class that admits solutions that solve the vacuum Einstein equations
and another that does not. In the latter, we find a deformation to the
Schwarzschild metric that solves the modified field equations in the small
coupling approximation. We calculate the event horizon shift, the innermost
stable circular orbit shift, and corrections to gravitational waves, mapping
them to the parametrized post-Einsteinian framework.
