My book sets out to prove that three normals drawn to any parabola $y^2=4ax$ from a given point $(h,k)$ are real when $h>2a$. It gives the following proof:

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> **Proof:** When normals are real, then all the three roots of $am^3+m(2a-h)+k=0$ are real and in that case,
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> $m\_1^2+m\_2^2+m\_3^2>0\\\Rightarrow (m\_1+m\_2+m\_3)^2-2(m\_1m\_2+m\_2m\_3+m\_3m\_1)>0\\\Rightarrow (0)^2-\frac{2(2a-h)}{a}>0\\\Rightarrow h>2a$
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I don't see how $m\_1^2+m\_2^2+m\_3^2>0$ is the condition for real roots of $m$. I tried solving few *depressed* cubic equations with imaginary roots, and found that the above condition is satisfied (at least for the ones I tried), but I was unable to find any proof online.

So how do I prove this condition?