$\frac{|x^{2}+kx+1|}{|x^{2}+x+1|}<2$

$\left\{\begin{matrix}k\in \left(\frac{\left(x+1\right)^{2}}{x},-3x-2-\frac{3}{x}\right)\text{, }&x<0\\k\in \mathrm{R}\text{, }&x=0\\k\in \left(-3x-2-\frac{3}{x},\frac{\left(x+1\right)^{2}}{x}\right)\text{, }&x>0\end{matrix}\right.$