Prove that all roots of $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real. Here $a,b \in \mathbb R$, and $a \neq 0$.
This is one of the previous year problem of Regional Math Olympiad (India). I had a hard time solving it, so thought I'd better ask here.
- Some real roots are possible: when $a<0$, the equation has two of them.
- If one more coefficient was allowed to be arbitrary: $a x^4 + b x^3 + cx^2 + x + 1 = 0$, then the roots could be all real, since every quartic can be brought into such form by scaling