It seems pretty straightforward:
Reduce[RealAbs[(c x + x^2 + 1) (10 c^2 + 11 c x + 3 x^2 - 1)] <
RealAbs[(2 c^2 - 3 c x + x^2 + 1)^2] &&
x > 0 && -1 <= c < 0,
{c, x}]
The answer is in terms of Root objects. You can plot the set of solutions:
RegionPlot[
RealAbs[(c x + x^2 + 1) (10 c^2 + 11 c x + 3 x^2 - 1)] <
RealAbs[(2 c^2 - 3 c x + x^2 + 1)^2], {c, -1, 0}, {x, 0, 10}]
