The range of the function $f\left(x\right)=\frac{x^2+x+2}{x^2+x+1},x\in R$ is

$f\left(x\right)=\frac{(x^2+x+1)+1}{x^2+x+1}=1+\frac{1}{{(x+\frac{1}{2})}^2+\frac{3}{4}}$

$\therefore f\left(x\right)>1$

$f\left(x\right)$ is max when ${(x+\frac{1}{2})}^2+\frac{3}{4}$ is minimum

$\therefore f_{\max }=1+\frac{1}{\frac{3}{4}}=\frac{7}{3}$ when $x=\frac{-1}{2}$

Range $=(1,\frac{7}{3}]$