Francis is the fourth youngest Matthews child. When Flynn was born, 4 of his siblings left because they had graduated. Shortly after another one followed suit, and then and then another. After the last one left that left Francis as the oldest sibling who still lived in the house. This didn't go to his head thankfully, he just grew very protective of his three younger siblings.
When his Hogwarts letter arrived, he was overjoyed, but also worried. This meant he'd be leaving his siblings alone. Once at Hogwarts he discovered something about himself. He was a prankster. He caused a wake of trouble his first 3 years at Hogwarts. Until in his fourth year, he injured another student during one of his pranks. He was mortified, he never wanted to hurt anyone. He swore off pranking and became interested in healing. He took Healing class, but found that to be boring. He soon dropped the class and focused on his new dream. Becoming an auror. That said, he still has a basic idea of how to heal people, which is all he really wanted.
For the rest of his Hogwarts years, Francis was a stellar student. He was never a Prefect or Head boy but that said he still was very respectable. Upon graduating Hogwarts he took up a simple job busing tables so he could earn some money for his family. He's been reading up on DAtDA and practicing so he'll be even more qualified when he applies to be an auror.
He did eventually apply as an Auror. To his great surprise and enjoyment he was accepted. He moved out, and soon after started dating Mia Anderson. He's lost contact with his family slightly but he wants to bring that back.