theres a ran (on the Rif) at the beginning of the masecta who wants to explain shittas rashi as follows: there are two way of min hatorah fulfilling ones obligation regarding tashbisu, either through bittul or through bedika/biur. the rashi on the mishna, explains the ran, is referring to one who doensnt do bittul, therefore his bedikas chametz is fulfilling a din d’oraisa. (how the ran says that no bittul was done is based on the fact that the gemara introduces the din that habodek tzarich sh’yivatel in the nmameof r’ yehuda in the name of rav, which makes it a chiddush from the gemara and not from the mishan itself). however the gemara on 4a already assumes that one must do bittul, and so his bedika is d’rabbanan. however the truth is, the halacha that bedika is d’rabannan doesnt mean to say that it isnt min hatorah. what it means is that since one can fulfill his obligation simply through bittul, the obligation to do bedika in spite of that is d’rabannan. so in the case of the rented house, the obligation to do bedika is only d’rabannan because the renter can simply do bittul, but still, chazal obligation someone to do a bedika on this house, the question is now who.
hope this helps. either way, look at the ran inside. it becomes pretty necessary to understand like him throughout the whole first perek