<s>[INST] How to solve $x^6 - 6x + 5 > 0$? Is there any elementary way to solve (i.e. not involving derivatives etc.) $x^6 - 6x + 5 > 0$ ? I can divide it two times by $(x - 1)$ to get $x^6 - 6x + 5 = (x - 1)^2 (x^4 + 2 x^3 + 3 x^2 + 4 x + 5)$, but it is not very helpful, because there is still this $(x^4 + 2 x^3 + 3 x^2 + 4 x + 5)$ part and I don't know how to prove that this is greater than $0$ without using calculus. What I want to say is that $x^6 - 6x + 5$ looks very specific and I am wondering if there exist any simple, yet smart way to solve it.
 [/INST] $x^4+2x^3+3x^2+4x+5=x^2(x+1)^2+2x^2+4x+5=x^2(x+1)^2+2（x+1)^2+3>0$
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