= Solution
For $y\in\mathbb R^m$ and $\gamma\geq0$, define the <optimization Lagrangian>
$$
\mathcal L(x,y,\gamma)=c^Tx+y^T(b-Ax)+\gamma(x^TQx-1).
$$
The Lagrangian primal is
$$
\boxed{\inf_{x\in\mathbb R^n}\ \sup_{y\in\mathbb R^m,\,\gamma\geq0}
\mathcal L(x,y,\gamma).}
$$
For feasible $x$, the supremum is $c^Tx$: equality terms vanish and the nonpositive inequality term has maximum zero at $\gamma=0$. If $Ax\ne b$, scale $y$ in the direction $b-Ax$ to make the supremum infinite. If the inequality fails, send $\gamma\to\infty$. Thus the inner supremum is exactly the objective extended to $+\infty$ off the <feasible set>. This proves equivalence with the original problem, including its infeasible case.
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