\(\dfrac{x}{x^2-x+1}=2008\Rightarrow\dfrac{x^2-x+1}{x}=\dfrac{1}{2008}\left(x\ne0\right)\)
\(\Rightarrow x-1+\dfrac{1}{x}=\dfrac{1}{2008}\)
\(\Rightarrow x+\dfrac{1}{x}=\dfrac{1}{2008}+1=\dfrac{2009}{2008}\)
\(M=\dfrac{x^2}{x^4+x^2+1}=\dfrac{1}{x^2+1+\dfrac{1}{x^2}}=\dfrac{1}{\left(x^2+2.x.\dfrac{1}{x}+\dfrac{1}{x^2}\right)-1}\)
\(\Rightarrow M=\dfrac{1}{\left(x+\dfrac{1}{x}\right)^2-1}=\dfrac{1}{\left(\dfrac{2009}{2008}\right)^2-1}=\dfrac{2008^2}{2009^2-2008^2}\)
\(\Rightarrow M=\dfrac{2008^2}{\left(2009-2008\right)\left(2009+2008\right)}=\dfrac{2008^2}{4017}\)
\(N=\dfrac{x^2}{x^4-x^2+1}=\dfrac{1}{x^2-1+\dfrac{1}{x^2}}=\dfrac{1}{\left(x^2+2.x.\dfrac{1}{x}+\dfrac{1}{x^2}\right)-3}\)
\(\Rightarrow N=\dfrac{1}{\left(x+\dfrac{1}{x}\right)^2-3}=\dfrac{1}{\left(\dfrac{2009}{2008}\right)^2-3}=\dfrac{2008^2}{2009^2-3.2008^2}\)
\(\Rightarrow N=\dfrac{2008^2}{4036081-12096192}=\dfrac{2008^2}{-8060111}\)
