Prove that $e^{2x} (x^2-x+1) > x^2+x+1$ if $x>0$. //Proof// By Taylor expansion, we have $$\begin{split} e^{2x} (x^2-x+1) &> (1+2x + (2x)^2/2)(x^2-x+1) ~\\ &=(1+2x+2x^2)(x^2-x+1) ~\\ & =1+2x+2x^2 -x-2x^2 -2x^3 + x^2 + 2x^3 + 2x^4 ~\\ &=2x^4 + x^2 +x+1~\\ &>x^2 + x+1 \end{split}$$