$\frac{x^2-x}{x^3-x}-\frac{x^2-1}{x^2\+x}$$\stackrel{\text{simplify}}{\=}$$-\frac{x^2-x-1}{x\left(x\+1\right)}$

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$\frac{x^2-x}{x^3-x}-\frac{x^2-1}{x^2\+x}$$\stackrel{\text{normal}}{\to }$$-\frac{x^2-x-1}{x\left(x\+1\right)}$

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$\frac{x^2-x}{x^3-x}-\frac{x^2-1}{x^2\+x}$$\stackrel{\text{simplify steps}}{\=}$$\begin{array}{lll}& & \text{Let's simplify}\\ & & \frac{x^2-1\cdot x}{x^3-1\cdot x}+\frac{\left(\mathrm{-1}\right)\cdot \left(x^2-1\right)}{x^2+x}\\ \text{•}& & \text{Factor}\\ & & \frac{x\left(x-1\right)}{x\left(x-1\right)\left(x+1\right)}+\frac{\left(\mathrm{-1}\right)\cdot \left(x^2-1\right)}{x^2+x}\\ \text{•}& & \text{Cancel out factor of}x\left(x-1\right)\text{provided}x\left(x-1\right)\ne 0\\ & & \frac{1}{x+1}+\frac{\left(\mathrm{-1}\right)\cdot \left(x^2-1\right)}{x^2+x}\\ \text{•}& & \text{Factor}\\ & & \frac{1}{x+1}-\frac{\left(x-1\right)\left(x+1\right)}{x\left(x+1\right)}\\ \text{•}& & \text{Cancel out factor of}x+1\text{provided}x\ne \mathrm{-1}\\ & & \frac{1}{x+1}-\frac{x-1}{x}\\ \text{•}& & \text{Find fractions to get lowest common denominator of}x\left(x+1\right)\\ & & \frac{x}{x}\cdot \left(\frac{1}{x+1}\right)+\frac{x+1}{x+1}\cdot \left(-\frac{x-1}{x}\right)\\ \text{•}& & \text{Multiply}\\ & & \frac{x\cdot 1}{x\left(x+1\right)}+\frac{\left(x+1\right)\cdot \left(-x+1\right)}{x\left(x+1\right)}\\ \text{•}& & \text{Add fractions}\\ & & \frac{x+\left(x+1\right)\left(-x+1\right)}{x\left(x+1\right)}\\ \text{•}& & \text{Expand}\\ & & -\frac{x}{x+1}+\frac{1}{x+1}+\frac{1}{x\left(x+1\right)}\\ \text{•}& & \text{Find fractions to get lowest common denominator of}x\left(x+1\right)\\ & & \frac{x}{x}\cdot \left(-\frac{x}{x+1}\right)+\frac{x}{x}\cdot \frac{1}{x+1}+\frac{1}{1}\cdot \left(\frac{1}{x\left(x+1\right)}\right)\\ \text{•}& & \text{Multiply}\\ & & \frac{x\cdot \left(-x\right)}{x\left(x+1\right)}+\frac{x\cdot 1}{x\left(x+1\right)}+\frac{1\cdot 1}{x\left(x+1\right)}\\ \text{•}& & \text{Add fractions}\\ & & \frac{-x^2+x+1}{x\left(x+1\right)}\end{array}$