<s>[INST] Prove that the function $(12-6x+x^2) e^x - (12+6x+x^2)$ is positive on $(0, \infty)$ 
Prove that the function $(12-6x+x^2) e^x - (12+6x+x^2)$ is positive on $(0, \infty)$. 

I am unable to prove this.
 [/INST] Let $f(x)=x+\ln(x^2-6x+12)-\ln(x^2+6x+12)$.
Since $f'(x)=1+\frac{2x-6}{x^2-6x+12}-\frac{2x+6}{x^2+6x+12}=\frac{x^4}{(x^2-6x+12)(x^2+6x+12)}\geq0$, 
we obtain that $f(x)\geq f(0)=0$.
Done!
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