Lucian responded in a comment on your question with this:
Why does Paul misquote the OT? Because he lived in the Orient (as opposed to the Occident) centuries before (as opposed to centuries after) Jerome's Vulgate and his obsession with Hebraica veritas.
To expand on that:
Modern English Bibles tend to agree with Jerome's Hebraica veritas argument, and primarily use the Masoretic Text (supplementing it with the Septuagint). Where the two textual traditions disagree, modern English Bibles generally follow the Masoretic Text's reading.
Paul, the inspired New Testament authors, and Jesus himself chose instead to often follow the Septuagint's reading when it diverges from the Masoretic Text:
- In Matthew 21:16, Jesus quotes Psalm 8:2 saying "Out of the mouth of infants and nursing babies you have ordained praise". In the Septuagint, Psalm 8:2 also says "ordained praise". However the Masoretic Text says "ordained strength".
- 1 Peter 4:18 quotes the Septuagint Proverbs 11:31 as "If the truly righteous is scarcely saved, what will become of the ungodly and the sinner?". The Masoretic Text says, "If the righteous is repaid on earth, how much more the wicked and the sinner!"
- Hebrews 11:21 quotes the Septuagint Genesis 47:31 with Jacob bowing in worship over the head of his staff, while the Masoretic Text has Jacob bowing over the head of his bed.
- Hebrews 10:5-7 quotes the Septuagint Psalm 40:6-8 as "a body have you prepared for me", while the Masoretic Text reads as "you have given me an open ear".
- Acts 13:41 quotes the Septuagint Habakkuk 1:5 starting with "Look, you scoffers", while the Masoretic Text starts with "Look among the nations".
- Acts 7:42-43 quotes the Septuagint Amos 5:25-27 as "You took up the tent of Moloch and the star of your god Rephan", while the Masoretic Text reads as "You shall take up Sikkuth your king, and Kiyyun your star-god".
- Acts 8:32-33 quotes the Septuagint Isaiah 53:7-8 as "In his humiliation justice was denied him. Who can describe his generation? For his life is taken away from the earth.", while the Masoretic Text reads as "By oppression and judgment he was taken away; and as for his generation, who considered that he was cut off out of the land of the living".
- James 4:6 quotes the Septuagint Proverbs 3:34 as "opposes the proud but gives grace to the humble", while the Masoretic Text reads similarly but with different underlying word-concepts as "scorns the scornful but he gives grace to the lowly".
It is noteworthy that Jerome's confidence in the "Hebraica veritas" caused him to issue a challenge in Apology Against Rufinus, Book II, Section 34:
"And further, I give a challenge to my accuser. I have shown that many things are set down in the New Testament as coming from the older books, which are not to be found in the Septuagint; and I have pointed out that these exist in the Hebrew. Now let him show that there is anything in the New Testament which comes from the Septuagint but which is not found in the Hebrew, and our controversy is at an end."
As noted earlier with numerous examples, that challenge has been met - and yet the controversy is still not at an end, and the Masoretic Text is still held above the Septuagint in the West.
Further information and references available here.
