Please refer to the attached question. Thanks.
Let $\displaystyle \frac{2x^2+5x+2}{2x^2-x+2}=y\Rightarrow (2-2y)x^2+(y+5)x+2-2y=0$
$\displaystyle x\in\mathbf{R}\Rightarrow\Delta\geq 0$
$\displaystyle \Delta=b^2-4ac=-15y^2+42y+9\geq 0$
$\displaystyle y_1=-\frac{1}{5}, \ y_2=3\Rightarrow y\in\left[-\frac{1}{5},3\right]$