I am certain that the other participants on the forum will recount the ecclesiastical history of this decision; suffice to say here that the Council of Trent definitely ratified the Catholic canon (we can debate how well defined it was prior to Trent, but that is not a discussion I wish to pursue here.) However, please note that Eastern Catholic Churches, embrace a broader canon.
I want to concentrate on the source text problems with adopting the Septuagint. There is no well defined boundary for the Septuagint. The traditional “inspired” Septuagint described in the Letter of Aristeas is only the Pentateuch.
The exact canon of the Septuagint is hotly debated, even among different Orthodox Churches in communion with each other: thus some include the Psalms of Solomon, some include 4 Esdras (= Protestant 2 Esdras = Slavonic 3 Esdras), some include 4 Maccabees; while others do not. The Slavonic and Greek Bibles do not exactly agree.
Worse, we have substantial differences in source manuscripts – and the differences are far greater than those among different NT manuscripts. Thus the scholarly Oxford New English Translation of the Septuagint has multiple translations reflecting the different literary translations (which are not just differing in minor details, but which differ dramatically). As one reviewer wrote:
Now to the truly extraordinary aspects of this new translation of the Septuagint. Students of the Septuagint will be aware that several of the Biblical books are known in two versions (in whole or in part) in Greek, with the divergent texts being published and annotated in both the Rahlfs handbook edition and in the Göttingen Septuaginta series. Sometimes the Old Greek version is paired with the version by Theodotion, as in the case of Susanna, Daniel, and Bel et Draco. Other times, the divergent texts of the great pandects Alexandrinus and Vaticanus are included, as in the case of Joshua. In the NETS, all the various divergent versions are translated. The usefulness for students of this act of simultaneous scholarly integrity and translational generosity can hardly be exaggerated. The books provided with multiple translations are: Iesous (Alexandrinus and Vaticanus for several chapters), Judges (again, A and B), 1-4 Reigns (Old Greek and Kaige), Esther (Old Greek and Alpha), Tobit (GII/Sinaiticus and GI/Alexandrinus & Vaticanus), Habbakoum 3 (majority text and the Barberini text), and Sousanna, Daniel, and Bel and the Dragon (Old Greek and Theodotion). In Job, the Theodotionic additions from Origen’s Hexaplaric text are included in brackets. In Sirach numerous verses from the Greek II text (not the Hebrew) are included in brackets. There may even be a few more passages, here and there, but those are the ones I was looking for and I’m very happy to find all of them translated. Such completeness needs to be applauded.
Finally, the Septuagint often disagrees with the Hebrew Masoretic text. While in some cases an eclectic text is useful (e.g., as developed in the RSV and especially the NRSV), it is also useful to study the Septuagint and Masoretic text independently, instead of forcing a harmonization that can never be definitive.
If you are seriously interested in the Septuagint, then I strongly recommend the New English Translation of the Septuagint. It does not contain an imprimatur (and it probably never could, since the Catholic Church does not recognize the canonicity of some of the included books) but it is a well-respected scholarly translation.