take cuttings having 3-5 nodes. & only 1 leaf at tip (If leaves are "large" cut leaf in half... then apply 0.1% IBA (rooting hormone) stick in well drained soil & mist

Begonias are propagated by differing methods depending on the type. Do you know what type of begonia you have? You could post a picture to the name that plant forum, or find it visually through a google search on the "images" tab. Finding the exact cultivar is not important to know how to proceed with your broken pieces, just which type. There is also a begonia forum on which your question has probably already been asked and answered.