Help to simplify this complicated fraction

I require help to simplify this. I used the method to make the denominator a single fraction then multiply the top by the reciprocal but when it comes to cancelling, I'm not sure if i've done it right.

$\frac{\frac{1}{{x}^{2}+x+2}}{1+\frac{2}{x({x}^{2}+x+2)}}$

I require help to simplify this. I used the method to make the denominator a single fraction then multiply the top by the reciprocal but when it comes to cancelling, I'm not sure if i've done it right.

$\frac{\frac{1}{{x}^{2}+x+2}}{1+\frac{2}{x({x}^{2}+x+2)}}$