Relation between inner product and norm in case of real matrix having negative real part in the eigenvalue

Question

I am trying to prove that if $A$ is a real matrix whose eigenvalues have all negative real part, then there exists a $b>0$ such that for all $x\in\mathbb{R}^n$, then $\langle Ax,x\rangle<-b \|x\|^2$. I started with the definition that $\langle Ax,x\rangle=\langle(-m+in)x,x\rangle= (-m+in)\langle x,x\rangle=(-m+in)\|x\|^2$. After that I am stuck. Can I use the idea of maximum eigenvalue? But since the eigenvalues can be complex, how can I compare them?