\(\frac{x}{x^2-x+1}=2008\Rightarrow\frac{x^2-x+1}{x}=\frac{1}{2008}\)
\(\Rightarrow\frac{x^2-x+1}{x}+2=\frac{1}{2008}+2\Rightarrow\frac{x^2+x+1}{x}=\frac{4017}{2008}\)
\(\Rightarrow\frac{x}{x^2+x+1}=\frac{2008}{4017}\)
\(A=\frac{x^2}{x^2+x^2+1}=\frac{x^2}{\left(x^2+1\right)^2-x^2}=\left(\frac{x}{x^2-x+1}\right)\left(\frac{x}{x^2+x+1}\right)=2008.\frac{4017}{2008}=4017\)