Ali wanting to marry Abu Jahls Daughter

Narrated Al-Miswar bin Makhrama:
'Ali demanded the hand of the daughter of Abu Jahl. Fatima heard of this and went to Allah's Apostle saying, "Your people think that you do not become angry for the sake of your daughters as 'Ali is now going to marry the daughter of Abu Jahl. "On that Allah's Apostle got up and after his recitation of Tashah-hud. I heard him saying, "Then after! I married one of my daughters to Abu Al-'As bin Al-Rabi' (the husband of Zainab, the daughter of the Prophet ) before Islam and he proved truthful in whatever he said to me. No doubt, Fatima is a part of me, I hate to see her being troubled. By Allah, the daughter of Allah's Apostle and the daughter of Allah's Enemy cannot be the wives of one man." So 'Ali gave up that engagement.
'Al-Miswar further said: I heard the Prophet talking and he mentioned a son-in-law of his belonging to the tribe of Bani 'Abd-Shams. He highly praised him concerning that relationship and said (whenever) he spoke to me, he spoke the truth, and whenever he promised me, he fulfilled his promise." (Bukhari Volume: 5, Book Number: 57, Hadith Number: 76)

Narrated Al-Miswar bin Makhrama:
I heard Allah's Apostle who was on the pulpit, saying, "Banu Hisham bin Al-Mughira have requested me to allow them to marry their daughter to Ali bin Abu Talib, but I don't give permission, and will not give permission unless 'Ali bin Abi Talib divorces my daughter in order to marry their daughter, because Fatima is a part of my body, and I hate what she hates to see, and what hurts her, hurts me." (bukhari Volume: 7, Book Number: 62, Hadith Number: 157)

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notes-- This hadith talks about the incident of `Ali wanting to marry the daughter of Abu Jahl, the bitterest enemy of Allah and His Messenger. Abu Jahl remained a kaafir 'til his death. He humiliated the Prophet and his Companions and tortured them; he also attempted the life of the Prophet many times.

Abu Jahl was finally killed by the Believers in the Battle of Badr long before `Ali proposed to marry his daughter. Thus, it becomes evident that the Prophet's justification of his disapproval of that marriage was not because he discouraged polygyny, as he himself was the best of examples and had multiple wives, but because of the shameful family history of the bride, which would definitely trouble his beloved daughter.