We have the equation
If there were two distinct real roots of the right hand side of $(1)$, then the minimum of the quadratic would be at their average and would be less than $0$; this because the positive coefficient of $x^2$ yields strict convexity. However, if there were an $x$ so that the right hand side were less than $0$, the left hand side would yield a sum of squares that was negative.
$\Rightarrow$ the roots cannot be real and distinct
The only possibility for a real root would be in the case where the left hand side of $(1)$ were $0$. for that to occur, each term in the sum would need to be $0$. That is, for all $0\le i\le n$,
$\Rightarrow$ $a$ and $b$ must be proportional.
If the roots of the right hand side of $(1)$ are equal or non-real, we have, from the quadratic formula, that
which upon rearrangement is Cauchy-Schwarz.
$\Rightarrow$ we want the roots of the right hand side of $(1)$ to be equal or non-real.