Begonias are easy to propagate, whether you use leaf or rhizome cuttings or you propagate by seed.
Always begin with a good medium for propagating, such as a mix of one-half perlite to one-half peat, and pack it down in a broad plastic container to create a flat, even surface. Sprinkle water on the medium, but don't saturate -- the soil should be evenly moist throughout.
1. The first method is simply to take a leaf cutting, treat the tip of the leaf petiole with rooting hormone, and plant the stem in the soil. You can also roll the leaf into a funnel shape before planting it. This keeps the humidity of the leaf cutting stable.
2. Another method of propagation is to make perpendicular slits in the major veins of the leaf and lay it down flat on the medium, firmly planting the hormone-treated stem of the leaf in the soil. Place small rocks on the leaf (not over the slits) to keep it anchored. This will result in several new plants growing from one leaf. Dividing one leaf up the middle of the stem and planting the two halves will also produce new plants.
To propagate using a rhizome cutting, sever a one- to three-inch piece with at least one leaf attached from the parent plant with a razor. Nestle the cutting in the moistened medium.
After planting your rhizome and leaf cuttings, mist them lightly with water and cover with plastic wrap. Keep the soil moist, and they will root in about six weeks.