Mattew: Herod died and his son Herod Archelaus (4 BC) became king after Jesus was born.
Luke: Quirinius was after Archelaus 6 Ad and later
also I d like to discuss with you prophecies about jesus in old testament, made a list and other stuff
Reply: The Problem of Luke 2:2 "This was the first census taken when Quirinius was governor of Syria"
One of the greatest difficulties in the Bible, in terms of its accuracy, is the census mentioned in Luke 2:2—a census that purportedly led Joseph and Mary to Bethlehem, where Jesus would be born. In conclusion, facile solutions do not come naturally to Luke 2:2. This does not, of course, mean that Luke erred. In agreement with Schürmann, Marshall “warns against too easy acceptance of the conclusion that Luke has gone astray here; only the discovery of new historical evidence can lead to a solution of the problem.”
Matthew and Luke have different accounts of the birth of Jesus. In fact, the accounts are so different that the only things Matthew and Luke agree on are that Mary and Joseph lived in Nazareth and that Jesus was born in Bethlehem. The rest of the information is different.
The two gospel writers record different things because they were writing to two different audiences. Matthew was a Jew who wrote to the Jews about Jesus, a Jew. Luke was the only Gentile writer of the entire Bible. He was a physician who never met Jesus. He learned all he knew from the other gospel writers.
Matthew and Luke both recorded valuable information. Do not think their information is in conflict with the other. Neither is wrong. They shared information from their own perspective.
The events didn't happen at the same time or in the same place. For instance, the shepherds and wise men did not show up at the birth of Jesus at the same time. Also, there were more than three wise men.
Matthew traces Jesus' roots back to Abraham to emphasize his role as the Jewish Messiah.
Matthew doesn't record the angel speaking to Mary as Luke does. On the other hand, Luke doesn't record the angel speaking to Joseph in a dream. Matthew did so mainly because Matthew had a Jewish audience. Joseph was faithful to the Jewish law and was considering divorcing Mary quietly instead of exposing her to public disgrace. Therefore, an angel appeared to him in a dream and told Joseph some things the angel did not tell Mary, according to Luke's gospel.
The angel told Joseph, "Joseph son of David, do not be afraid to take Mary home as your wife, because what is conceived in her is from the Holy Spirit. She will give birth to a son, and you are to give him the name Jesus because he will save his people from their sins.”
Matthew, like Luke, does say Jesus was born in Bethlehem. However, Matthew does not record the journey to Bethlehem or give the reason Joseph had to go when Mary was about to give birth. We find out those details in the Gospel of Luke.
Matthew doesn't mention there not being any room in the inn or a manger. After telling about the angel speaking to Joseph, Matthew jumps right into telling the story about the Magi, or wise men. He gives more details about them than he does about the birth of Jesus which includes Mary and Joseph fleeing from Bethlehem to Egypt to escape Herod. If you read the story carefully, you will see that Jesus is no longer a baby when the wise men arrived with their gifts. It took them about two years for their caravan to get to Bethlehem after stopping in Jerusalem. Jesus was no longer a baby, and His parents were living in a house at that time.
Matthew doesn't record the journey to Bethlehem at all, but he records them leaving the city and going back to Nazareth. It is surprising that Matthew didn't record how Mary and Joseph went to Jerusalem so Jesus could be circumcised when He was eight days old because it was a Jewish law.
What we know about the birth of Jesus comes mostly from Luke 2:1-39. Luke gives much more information than Matthew. Luke traces Jesus' ancestry all the way back to Adam. Luke says His family lived in Nazareth and went to Bethlehem for a census required by Caesar Augusta that all the world should be taxed in their own city. Matthew didn't give us that information.
When the baby was born, Mary wrapped him in swaddling clothes and laid him in a manger because there was no room for them in the inn. Matthew left that out of his gospel.
Shepherds were abiding in the field, keeping watch over their flock by night when an angel of the Lord came upon them and told them the good news that in the city of David a Savior was born. Then the angel gave the shepherd a sign that they would find the babe lying in a manger and wrapped in swaddling clothes. After the angel left, the shepherds went to Bethlehem to see what had come to pass. They found Mary, Joseph, and the baby lying in a manger. The shepherds returned to their fields glorifying and praising God for all the things that they had heard and seen. It was just as the angel told them.
Matthew tells what was important to him and his audience. Luke tells some of the same stories from his perspective.
Matthew has the dream of Joseph. Luke has the Annunciation to Mary.
Matthew told about the flight to Egypt. Luke told about the journey to Bethlehem and being there.
Matthew tells the story of the wise men. Luke tells the story of the shepherds.
Matthew and Luke give Joseph’s genealogy but in different ways. Matthew gives Jesus' genealogy only back to Abraham, the first Jew. Luke gives the genealogy as far back as he could. He traces Jesus' ancestry back to Adam, the first man God created.
Matthew doesn't show readers what happened after Jesus was born in Bethlehem. Unless you read the Gospel of Luke, you would not know that there was no room for Mary and Joseph and Jesus was born in a stable, wrapped in swaddling cloths and placed in a manger. Matthew was silent about those things.