\(\dfrac{1}{x^2+x+1}-\dfrac{1}{x-x^2}-\dfrac{x^2+2x}{x^3-1}\)
\(=\dfrac{\left(x-1\right)x}{x\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{x^2+x+1}{x\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{\left(x^2+2x\right)x}{x\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{x^2-x+x^2+x+1-x^3-2x^2}{x\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{1-x^3}{x\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{-\left(x^3-1\right)}{x\left(x^3-1\right)}=\dfrac{-1}{x}\)
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