The class q-Horn, introduced by Boros, Crama and Hammer in 1990, is one of the largest known classes of propositional CNF fomulas for which satisfiability decision is polynomial. This class properly contains the fundamental classes of Horn and 2CNF formulas as well as the class of renamable (or disguised) Horn formulas. In this paper we gradually extend this class such that its favorable algorithmic properties can be made accessible to formulas that are outside but ``close'' to this class. We show that satisfiability decision is fixed-parameter tractable parameterized by the distance of the given formula from q-Horn. The distance is measured as the smallest number of variables that we need to delete from F in order to get a q-Horn formula, i.e., the size of a smallest deletion backdoor set into the class q-Horn. This result generalizes known fixed-parameter tractability results for satisfiability decision with respect to the parameters distance from Horn or 2CNF (Nishimura, Ragde, Szeider 2004), and with respect to renamable Horn (Razgon, O'Sullivan 2009).