<s>[INST] How to derive the condition? (I am not sure it is correct.) Given a polynomial with real coefficients $\alpha x^2 + \beta x + a^2 + b^2 + c^2 - ab- bc - ca $ has imaginary roots, how do we prove $ 2(\alpha - \beta) + ((a - b)^2 + (b - c)^2 + (c - a)^2) > 0 $?
Given all coefficients are real, roots are $ z $ and $\overline{z}$. Hence, $z \overline{z} > 0$. Hence, $ \dfrac{a^2 + b^2 + c^2 - ab - bc - ca}{\alpha} > 0 $. Thus $ \alpha > 0$ because numerator is greater than $ 0 $. Now $D < 0$. Therefore,
$ \begin{align}
\beta^2 - 2 \alpha ((a - b)^2 + (b - c)^2 + (c - a)^2) < 0\\
\end{align}$ 
How to proceed further?
 [/INST] You can try with $2f(-1)>0$
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