Am I real?

Algebra Level 3

Let \(a,b,c \in \mathbb R^+\) be distinct numbers such that \(a+b+c=1\).

Let \(\alpha = \text{min}(a^3 +a^2bc, b^3 + b^2ca, c^3 + c^2ab)\)

Then, what can you say about the roots of the equation

\[x^2 + x + 4\alpha = 0\]

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