Under the assumptions that p and q are regularly varying functions satisfying conditions∫ ∞ a t p(t) 1α dt < ∞ and ∫ ∞ a ( t p(t) ) 1 α dt = ∞ existence and asymptotic form of regularly varying intermediate solutions are studied for a fourth-order quasilinear differential equation( p(t)|x′′(t)|α−1 x′′(t) )′′ + q(t)|x(t)|β−1 x(t) = 0, α > β > 0. It is shown that under certain integral conditions there exist two types of intermediate solutions which according to their asymptotic behavior is to be divided into six mutual distinctive classes, while asymptotic behavior of each member of any of these classes is governed by a unique explicit law,