If x(≠0) $$\epsilon R$$, then $$\frac{1}{1 + x + x^2} + \frac{1}{1 + x + x^{-1}} + \frac{1}{1 + x^{-1} + x^{-2}} =$$

Retain the first term as it is. 

Multiply the numerator and denominator of the second term by x. 
it gets transformed to $$\frac{x}{x+x^2+1}$$

Multiply the numerator and denominator of the third term by x*x
It gets transformed to $$\frac{x^2}{x+x^2+1}$$

As can be seen, the denominators of all three terms are equal. 
The numerators can all be added up directly now. It is seen that numerator and denominator are equal now.
Hence, answer = 1 i.e. option C