Reducible and δ-reducible handle additions

Let M be α simple 3-manifold with F α component of δM of genus at least two. For a slope α on F, we denote by M(α) the manifold obtained by attaching α 2-handle to M along a regular neighborhood of α on F. Suppose that α and β are two separating slopes on F such that M(α) and M(β) are reducible. Then the distance between α and β is at most 2. As a corollary, if g(F) = 2, then there is at most one separating slope γ on F such that M(γ) is either reducible or δ-reducible.