Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots +a_n$ cannot be completely resolved into linear factors if $a_1^2
Prove that a real polynomial $x^n+ a_1x^{n-1}+ \cdots +a_n$ cannot be completely resolved into linear factors if $a_1^2<a_2$.
Here's what I've got. Let $\alpha_1, \dots, \alpha_n$ be the roots of the polynomial. Then $\displaystyle{a_1^2-a_2=(\alpha_1+ \cdots +\alpha_n)^2 - \sum_{k,j \leq n; k \neq j}\alpha_{j}\alpha_{k} = \alpha_1^2+ \cdots +\alpha_n^2 + \sum_{k,j \leq n; k \neq j}\alpha_{j}\alpha_{k}}$, but I have no idea how to prove this nonnegative. Please help.

Assume that $\alpha_1,...,\alpha_n$ are $n$ real roots of the polynomial. Notice that $$\sum_{i<j} \alpha_i \alpha_j = a_2 > a_1^2 \geq 0\implies a_1^2-a_2 = \sum_i \alpha_i^2+\sum_{i<j}\alpha_i \alpha_j>0$$
which is a contradiction with our assumption that $a_1^2-a_2<0$. Hence, these real roots cannot exist at the same time.
