How did Pope Alex Borgia die?

Answer 1

His son Cesare Borgia must have killed him because they were locked in a room together after being robbed and when the guards opened the door Rodrigo was dead from poison. Cesare was famous for poisoning dinner guests and he even poisoned his brother.

Answer 2

I hope you are referring to Rodrigo de Borja who was elected pope as Alexander VI as there is no pope named Alex Borgia.
He and his son Cesare had dined with a cardinal one evening. Shortly after, both became quite ill. Cesare survive but the Pope did not. Here is the description of his death:
Saturday, the 12th of August, 1503, the Pope fell ill in the morning. After the hour of vespers, between six and seven o'clock a fever appeared and remained permanently. On the 15th of August thirteen ounces of blood were drawn from him and the tertian ague supervened. On Thursday, the 17th of August, at nine o'clock in the forenoon he took medicine. On Friday, the 18th, between nine and ten o'clock he confessed to the Bishop Gamboa of Carignola, who then read Mass to him. After his Communion he gave the Eucharist to the Pope who was sitting in bed. Then he ended the Mass at which were present five cardinals, Serra, Juan and Francesco Borgia, Casanova and Loris. The Pope told them that he felt very bad. At the hour of vespers after Gamboa had given him Extreme Unction, he died.
Both were apparently poisoned but there is no proof.

His body started to decompose almost immediatey after his death and is described by a witness below:

"It was a revolting scene to look at that deformed, blackened corpse, prodigiously swelled, and exhaling an infectious smell; his lips and nose were covered with brown drivel, his mouth was opened very widely, and his tongue, inflated by poison, fell out upon his chin; therefore no fanatic or devotee dared to kiss his feet or hands, as custom would have required." The Venetian ambassador stated that the body was "the ugliest, most monstrous and horrible dead body that was ever seen, without any form or likeness of humanity."

Answer 3

Pope Alexander died in Rome, either by poisoning or due to malaria that was prevalent in the area at the time.

Answer 4

Pope Alexander II died on 1073-04-21.