Using the method of Section 8.7.2 construct the set of cubic interactions of a massless spin-2 field embedded in hμν. There are many terms, all with two derivatives, but their coefficients are precisely fixed. You can also check that this is the same thing you get from expanding MPl2ημν+MPl1hμνR[ημν+MPl1hμν] to cubic order in hμν.
It should be clear that the same method will produce the terms fourth order in hμν, however, these are suppressed by MPl21. Most tests of general relativity probe only that it is described by a minimally coupled spin-2 field (e.g.\ bending of light, gravitational waves, frame dragging). Some precision tests assay the cubic interactions (e.g.\ the perihelion shift of Mercury). No experiment has yet tested the quartic interactions.