a ) \(\dfrac{5x+1}{x^3-1}-\dfrac{1-2x}{x^2+x+1}-\dfrac{2}{1-x}\)
\(=\dfrac{5x+1}{x^3-1}-\dfrac{\left(1-2x\right)\left(x-1\right)}{x^3-1}+\dfrac{2}{x-1}\)
\(=\dfrac{5x+1-\left(1-2x\right)\left(x-1\right)+2\left(x^2+x+1\right)}{x^3-1}\)
\(=\dfrac{5x+1-\left(x-2x^2-1+2x\right)+2x^2+2x+2}{x^3-1}\)
\(=\dfrac{5x+1-\left(-2x^2+3x-1\right)+2x^2+2x+2}{x^3-1}\)
\(=\dfrac{5x+1+2x^2-3x+1+2x^2+2x+2}{x^3-1}\)
\(=\dfrac{4x^2+4x+4}{x^3-1}=\dfrac{4\left(x^2+x+1\right)}{\left(x-1\right)\left(x^2+x+1\right)}\)
\(=\dfrac{4}{x-1}\)
b ) \(\dfrac{5}{x+1}-\dfrac{10}{x-x^2-1}-\dfrac{15}{x^3+1}\)
\(=\dfrac{5\left(x^2-x+1\right)}{x^3+1}+\dfrac{10}{x^2-x+1}-\dfrac{15}{x^3+1}\)
\(=\dfrac{5\left(x^2-x+1\right)}{x^3+1}+\dfrac{10\left(x+1\right)}{x^3+1}-\dfrac{15}{x^3+1}\)
\(=\dfrac{5x^2-5x+5+10x+10-15}{x^3+1}\)
\(=\dfrac{5x^2+5x}{x^3+1}\)
\(=\dfrac{5x\left(x+1\right)}{\left(x+1\right)\left(x^2-x+1\right)}=\dfrac{5x}{x^2-x+1}\)
