<s>[INST] How can one prove that this polynomial is non-negative? How one can prove the following inequality?
$$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74\geq 0$$ 
I plotted the graph on Wolfram Alpha and found that the inequality seems to hold. I was unable to represent the polynomial as a sum of squares. 
It looks quite boring to approximate the derivative to be zero and use some numerical methods to show that values near local minimums proves that the inequality really holds everywhere.
 [/INST] For $x<0$ it's obvious.
But for $x\geq0$ we obtain:
$$58x^{10}-42x^9+11x^8+42x^7+53x^6-160x^5+118x^4+22x^3-56x^2-20x+74=$$
$$=(x^3-x^2-x+1)(58x^7+16x^6+85x^5+85x^4+207x^3+47x^2)+$$
$$+287x^4-138x^3-103x^2-20x+74>0,$$
where $$287x^4-138x^3-103x^2-20x+74=$$
$$=(16x^2-4x-5)^2+(31x^4-10x^3+x^2)+(40x^2-60x+49)>0.$$
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