>I also did not know that the sum of
>squares and square of sum of polynomial roots were related
>to the coefficients in any meaningful way.
They are, and this link is the crux of the proof. The link is the elementary symmetric polynomials. Before I can explain this I shall need to set up some notation. For any term t in variables x_1, x_2, ... x_n,
I shall use ssum(t) to mean the sum of all terms which may be obtained from t by permuting the variables. So for example, ssum((x_1)^2) = (x_1)^2 + (x_2)^2 + ... + (x_n)^2.

We know that P(x) has the form a (x - x_1)(x- x_2)...(x - x_n). Multiplying out, we obtain b = a ssum(x_1), c = a ssum(x_1*x_2). The other coefficients are similarly expressed in terms of polynomials in the x_i; the elementary symmetric polynomials.

Now (ssum(x_1))^2 = ssum((x_1)^2) + 2 ssum(x_1 x_2), so ssum((x_1)^2) = (ssum(x_1))^2 - 2 ssum(x_1 x_2) = (b/a)^2 - c/a = (b^2 - 2ac)/a^2, as stated in the solution you ask about.

Thankyou

sfwc
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