If     and     are extreme points of  , then

 

The given function is $f\left(x\right)=\alpha \log \left|x\right|+\beta x^2+x$

Differentiate both sides 

$f\text{'}\left(x\right)=\frac{\alpha }{x}+2\beta x+1$

it has roots $x=-1, and x=2$

$$\begin{gathered} f\text{'}\left(-1\right)=0 \\ -\alpha -2\beta +1=0 \end{gathered}$$

and 

$$\begin{gathered} f\text{'}\left(2\right)=0 \\ \frac{\alpha }{2}+4\beta +1=0 \end{gathered}$$

Therefore, $\alpha =2,\beta =-\frac{1}{2}$