M.A. Sychev has recently shown that conditions necessary and sufficient for the lower semicontinuity of integral functionals with p-coercive integrands are W1,p-quasi-convexity and the so-called matching condition (M). Condition (M) is so general that there is the conjecture that is always holds in the case of continuous integrands. The paper develops relaxation theory (construction of lower semicontinuous envelopes) under the assumption that condition (M) holds. It turns out that, in this case, the theory has very good structure. Applications of general relaxation theory to particular cases, including the theory of strong materials, are also discussed.