From a study of broad absorption line (BAL) + infrared (IR) + FeII quasi-stellar objects (QSOs) [using deep Gemini Multi-Object Spectrograph Integral Field Unit (GMOS-IFU) spectroscopy], new results are presented for IRAS 04505-2958. Specifically, we have studied in detail the outflow (OF) process at two large galactic scales: (i) two blobs/shells at radius r ~ 1.1 and 2.2 kpc, and (ii) an external hypergiant shell at r ~ 11 kpc. In addition, the presence of two very extended hypergiant shells at r ~ 60-80 kpc is also discussed. From this GMOS study the following main results were obtained. (i) For the external hypergiant shell, the kinematics GMOS maps of the ionized gas ([OII], [NeIII], [OIII], Hβ) show a small-scale bipolar OF, with similar properties to those observed in the prototype of exploding external supershells: NGC 5514. (ii) Three main knots - of this hypershell S3 - show the presence of a young starburst. (iii) The two internal shells show OF components with typical properties of nuclear shells. (iv) The two blobs and the hypershell are aligned at PA ~ 131° showing bipolar OF shape at ~10-15 kpc scale. In addition, the more external shells (at ~60-80 kpc scale) are aligned at PA ~ 40° also with bipolar OF shape (perpendicular to the more internal OF). (v) A strong blue continuum and multiple emission-line components were detected in all the GMOS fields. The new GMOS data show a good agreement with an extreme + explosive OF scenario for IRAS 04505-2958, in which part of the interstellar medium (ISM) of the host galaxy was ejected (in multiple shells). This extreme OF could also be associated with two main processes in the evolution of QSOs: (i) the formation of companion/satellite galaxies by giant explosions; and (ii) to define the final mass of the host galaxy, and even if the explosive nuclear OF is extremely energetic, this process could disrupt an important fraction of the host galaxy. Finally, the generation of ultra-high-energy cosmic rays and neutrino/dark matter - associated with HyNe in explosive BAL + IR + FeII QSOs - is discussed.