[Enlil-An]

While Matthew might not specifically say that Mary and Joseph are from Bethlehem, a close reading of the story shows that this is so. There's no mention of the census or any trip from Nazareth. After Jesus is born, they're still in Bethlehem two years later when the wise men find them and the only reason they leave is because Herod threatens Jesus' life. After Herod's death, they left Egypt, but Matthew makes it clear the only reason they didn't return to Bethlehem (home) was because Herod's son was ruler in Judea, so Joseph took his family and "he turned aside into the parts of Galilee: And he came and dwelt in a city called Nazareth".

Luke's gospel says that Mary and Joseph travelled from Nazareth to Bethlehem, staid there a month (not two years) and then immediately returned to Nazareth after presenting the baby Jesus to the temple.

Herod's decree, the wisemen, the star, the flight to Egypt; are all from Matthew's gospel. These elements of the story do not appear in Luke's. Likewise, the census, the trip to Bethlehem, the manger story, the shepards; these all appear in Luke but not Matthew. The only things that both accounts agree on is the virgin birth and that Jesus was born at Bethlehem. Other than that, they are completely different stories.