<s>[INST] All roots of the quartic equation $a x^4 + b x^3 + x^2 + x + 1 = 0$ cannot be real Problem

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 $a  \neq  0$.

Source
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.
Observations

*

*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

 [/INST] Let $f(x) = x^4 + x^3 + x^2 + bx + a$.  Then the quartic
$ax^4 + bx^3 + x^2 + x + 1$ is $x^4 f(1/x)$, and has four real roots
iff $f$ does.  But $f''(x) = 12x^2 + 6x + 2 > 0$ for all $x$.
Therefore $f$ is convex upwards and hence can have at most two real roots, QED.
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