\(\dfrac{5x+1}{x^3-1}-\dfrac{1-2x}{x^2+x+1}-\dfrac{2}{1-x}\)
\(=\dfrac{5x+1}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{1-2x}{x^2+x+1}+\dfrac{2}{x-1}\) MTC: \(\left(x-1\right)\left(x^2+x+1\right)\)
\(=\dfrac{5x+1}{\left(x-1\right)\left(x^2+x+1\right)}-\dfrac{\left(1-2x\right)\left(x-1\right)}{\left(x-1\right)\left(x^2+x+1\right)}+\dfrac{2\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{\left(5x+1\right)-\left(1-2x\right)\left(x-1\right)+2\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{\left(5x+1\right)-\left(x-1-2x^2+2x\right)+\left(2x^2+2x+2\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{5x+1-x+1+2x^2-2x+2x^2+2x+2}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{4x^2+4x+4}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{4\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{4}{x-1}\)
