Thank you sir for pointing out my mistake.I understood where I went wrong.Appreciate it.I did put the other expression into account but my first expansion itself is wrong.So,it ultimately become wrong for the last part.Would you mind explaining why is completing the square method invalid while the latter should? I seriously do want to know..Thanks :)

Hi, your completed square is wrong. The completed square for x^2+x+1 should be (x+1/2)^2 + 3/4. Also, if you use completing the square method, it would be more tedious expanding (1+x+x^2)^10 because you cannot use the shorter method of expanding (1+b)^n to expand ((x+1/2)^2 + 3/4)^10.

Oh that's how it is.Yup I agree on the tediousness part.Great thanks for the explanation. :)