  We consider a continuum model of active viscoelastic matter, whereby an
active nematic liquid-crystal is coupled to a minimal model of polymer dynamics
with a viscoelastic relaxation time $\tau_C$. To explore the resulting
interplay between active and polymeric dynamics, we first generalise a linear
stability analysis (from earlier studies without polymer) to derive criteria
for the onset of spontaneous heterogeneous flows (strain rate) and/or
deformations (strain). We find two modes of instability. The first is a viscous
mode, associated with strain rate perturbations. It dominates for relatively
small values of $\tau_C$ and is a simple generalisation of the instability
known previously without polymer. The second is an elastomeric mode, associated
with strain perturbations, which dominates at large $\tau_C$ and persists even
as $\tau_C\to\infty$. We explore the novel dynamical states to which these
instabilities lead by means of direct numerical simulations. These reveal
oscillatory shear-banded states in 1D, and activity-driven turbulence in 2D
even in the elastomeric limit $\tau_C\to\infty$. Adding polymer can also have
calming effects, increasing the net throughput of spontaneous flow along a
channel in a new type of "drag-reduction". Finally the effect of including
strong, antagonistic coupling between nematic and polymer is examined
numerically, revealing a rich array of spontaneously flowing states.
