Question 779018
x^2+x+1 cannot be factored because it is not a square trinomial.  Subtract 1 from both sides, and then decide what to add to x^2+x to make it a square factorable trinomial!  Study the derivation of Completing the Square.  The term to add is (1/2)^2=1/4.  


Knowing THAT, let's start from the original equation again and use the term.


{{{x^2+x+1=0}}}
{{{x^2+x =-1}}}
{{{x^2+x+1/4=-1+1/4}}}
{{{(x+1/2)^2=-1+1/4}}}  which you should finish on your own.