You have correctly differentiated to find:

\(\displaystyle f'(x)=\frac{1-x^2}{(x^2+x+1)^2}=\frac{(1+x)(1-x)}{(x^2+x+1)^2}\)

The denominator has no real roots, since the discriminant is negative. So, you only need test the sign of the numerator on the intervals:

\(\displaystyle (-\infty,-1),\,(-1,1),\,(1,\infty)\)

From this, you can find the nature of the two extrema.