jmc

We can write $(x+1{)}^{2010}=[(x+1{)}^2{]}^{1005}=(x^2+2x+1{)}^{1005}.$ This leaves the same remainder as $x^{1005}$ when divided by $x^2+x+1.$

Then $x^{1005}-1=(x^3{)}^{335}-1$ is divisible by $x^3-1=(x-1)(x^2+x+1).$ Therefore, the remainder when $(x+1{)}^{2010}$ is divided by $x^2+x+1$ is $\boxed{1}.$