Recently, Sychev showed that conditions both necessary and sufficient for lower semicontinuity of integral functionals with p-coercive extended-valued integrands are the W^{1,p}-quasi-convexity and the validity of a so-called matching condition (M). Condition (M) is so general that we conjecture whether it always holds in the case of continuous integrands. In this paper we develop the relaxation theory under the validity of condition (M). It turns out that a better relaxation theory is available in this case. This motivates our research since it is an important old open problem to develop the relaxation theory in the case of extended-value integrands. Then we discuss applications of the general relaxation theory to some concrete cases, in particular to the theory of strong materials.