Consider $x^2+ax+b=0$, where $x$ is the variable and $a,b$ are complex coefficients. Is there any condition on $a$ and $b$ which makes sure the roots of the equation have negative real parts?
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Let $\,z_1:=a+bi\,,\,z_2:=c+di\in\Bbb C\,$ be the two roots, then using Viete's formulae:
Now compare real and imaginary parts in the two expressions above.