Quickly checking if an inequality holds on a convex region

Question

Let $C$ be a given convex polygon in $\mathbb{R}^2$ containing the origin and let $a$, $\mathbf{b}$, and $Q\succeq0$ be a given scalar, vector, and matrix respectively. Is there a fast way to verify if the expression $$a+\mathbf{b}^T\mathbf{x}+\mathbf{x}^TQ\mathbf{x} - \|\mathbf{x}\|\geq 0$$ holds for all $\mathbf{x}\in C$? This function is not convex, but it seems that it ought to be simple enough to verify in some clever way.