Question

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

Answer 1

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

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

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

\(=1+\dfrac{-2x+1}{x^2+x+1}\cdot\dfrac{x}{2x-1}\)

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