  Four-body distorted wave theory appropriate for nucleon-nucleus reactions
leading to 3-body continuum excitations of two-neutron Borromean halo nuclei is
developed. The peculiarities of the halo bound state and 3-body continuum are
fully taken into account by using the method of hyperspherical harmonics. The
procedure is applied for A=6 test-bench nuclei; thus we report detailed studies
of inclusive cross sections for inelastic $^6$He(p,p')$^6$He$^*$ and
charge-exchange $^6$Li(n,p)$^6$He$^*$ reactions at nucleon energy 50 MeV. The
theoretical low-energy spectra exhibit two resonance-like structures. The first
(narrow) is the excitation of the well-known $2^+$ three-body resonance. The
second (broad) bump is a composition of overlapping soft modes of
multipolarities $1^-, 2^+, 1^+, 0^+$ whose relative weights depend on
transferred momentum and reaction type. Inelastic scattering is the most
selective tool for studying the soft dipole excitation mode.
