= Solution
There is \b[no such universal constant]. For a real parameter $M$, consider the real <matrix>
$$
B_M=\begin{pmatrix}1&M\\0&1\end{pmatrix},\qquad x=\begin{pmatrix}1\\1\end{pmatrix}.
$$
Its <characteristic polynomial> is $(t-1)^2$, so both <eigenvalues> are real and $\lambda_{\max}(B_M)=1$. But
$$
\frac{x^\dagger B_Mx}{x^\dagger x}=1+\frac M2.
$$
For any proposed finite $C$, choosing $M>2(C-1)$ violates the proposed bound. This is an explicit illustration that <nonnormal eigenvalues do not bound a quadratic quotient>.
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