  We consider the cubic-quintic nonlinear Schr\"odinger equation: \[
i\partial_t u = -\Delta u - |u|^2u + |u|^4u. \]
  In the first part of the paper, we analyze the one-parameter family of
ground-state solitons associated to this equation with particular attention to
the shape of the associated mass/energy curve. Additionally, we are able to
characterize the kernel of the linearized operator about such solitons and to
demonstrate that they occur as optimizers for a one-parameter family of
inequalities of Gagliardo--Nirenberg type. Building on this work, in the latter
part of the paper we prove that scattering holds for solutions belonging to the
region $\mathcal{R}$ of the mass/energy plane where the virial is positive. We
show this region is partially bounded by solitons but also by rescalings of
solitons (which are not soliton solutions in their own right). The discovery of
rescaled solitons in this context is new and highlights an unexpected
limitation of any virial-based methodology.
