<s>[INST] How do we prove $x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0$? Question
How do we prove the following for all $x \in \mathbb{R}$ :
$$x^6+x^5+4x^4-12x^3+4x^2+x+1\geq 0 $$
My Progress
We can factorise the left hand side of the desired inequality as follows:
$$x^6+x^5+4x^4-12x^3+4x^2+x+1=(x-1)^2(x^4+3x^3+9x^2+3x+1)$$
However, after this I was unable to make any further progress in deducing the desired inequality.
I appreciate your help
 [/INST] $$x^4+3x^3+9x^2+3x+1=\left(x^2+\frac32x+1\right)^2+\frac{19}4x^2$$ is always positive.
There is a general theorem that if a real polynomial is always non-negative, it can be written as a sum of squares.
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