Toroidal and boundary-reducing Dehn fillings

Let M be a simple 3-manifold with a toral boundary component ∂0M. If Dehn filling M along ∂0M one way produces a toroidal manifold, and Dehn filling M along ∂0M another way produces a boundary-reducible manifold, then we show that the absolute value of the intersection number on ∂0M of the two filling slopes is at most two. In the special case that the boundary-reducing filling is actually a solid torus and the intersection number between the filling slopes is two, more is said to describe the toroidal filling.