<s>[INST] Solving $x^8 - x^5 + x^2 - x + 1 > 0$ over $\mathbb{R}$ I could not find any decent approach to solve this inequality. I would appreciate any help, and input if this is even possible to solve(without a computer).
$$x^8-x^5+x^2-x+1>0$$
 [/INST] Using the AM-GM inequality, we have
$$x^8 + \frac{x^2}{2} \geqslant 2\sqrt{x^8 \cdot \frac{x^2}{2}} = \sqrt{2} \cdot |x^5| \geqslant x^5,$$
and
$$1+\frac{x^2}{2}  \geqslant 2\sqrt{\frac{x^2}{2}} = \sqrt 2 \cdot |x| \geqslant x$$
Therefore
$$x^8+x^2+1> x^5 + x.$$
Done.
Note. The SOS form
$$x^{8}+x^{2}+1-x^{5}-x=\left(x^{4}-\frac{x}{2}\right)^{2}+\left(\frac{x}{2}-1\right)^{2}+\frac{x^{2}}{2}>0.$$
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