5. \( \frac{x^{2}-x}{x^{2}-2 x-3} \div \frac{x^{2}-4 x+3}{1-x^{2}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(x^{2}-x\right)}{\left(x^{2}-2x-3\right)}\times \left(1-x^{2}\right)}{\left(x^{2}-4x+3\right)}\) - step1: Remove the parentheses: \(\frac{\frac{x^{2}-x}{x^{2}-2x-3}\times \left(1-x^{2}\right)}{x^{2}-4x+3}\) - step2: Multiply the terms: \(\frac{\frac{\left(x^{2}-x\right)\left(1-x\right)}{x-3}}{x^{2}-4x+3}\) - step3: Multiply by the reciprocal: \(\frac{\left(x^{2}-x\right)\left(1-x\right)}{x-3}\times \frac{1}{x^{2}-4x+3}\) - step4: Rewrite the expression: \(\frac{x\left(x-1\right)\left(1-x\right)}{x-3}\times \frac{1}{\left(x-1\right)\left(x-3\right)}\) - step5: Reduce the fraction: \(\frac{x\left(1-x\right)}{x-3}\times \frac{1}{x-3}\) - step6: Multiply the terms: \(\frac{x\left(1-x\right)}{\left(x-3\right)\left(x-3\right)}\) - step7: Multiply the terms: \(\frac{x\left(1-x\right)}{\left(x-3\right)^{2}}\) - step8: Multiply the terms: \(\frac{x-x^{2}}{x^{2}-6x+9}\) To solve the expression \[ \frac{x^{2}-x}{x^{2}-2x-3} \div \frac{x^{2}-4x+3}{1-x^{2}}, \] we can rewrite the division as multiplication by the reciprocal: \[ \frac{x^{2}-x}{x^{2}-2x-3} \times \frac{1-x^{2}}{x^{2}-4x+3}. \] Now, let's simplify each part step by step. 1. **Factor the expressions:** - \(x^2 - x = x(x - 1)\) - \(x^2 - 2x - 3 = (x - 3)(x + 1)\) - \(1 - x^2 = -(x^2 - 1) = -(x - 1)(x + 1)\) - \(x^2 - 4x + 3 = (x - 3)(x - 1)\) 2. **Substituting the factored forms into the expression:** \[ \frac{x(x - 1)}{(x - 3)(x + 1)} \times \frac{-(x - 1)(x + 1)}{(x - 3)(x - 1)}. \] 3. **Combine the fractions:** \[ \frac{x(x - 1)(-(x - 1)(x + 1))}{(x - 3)(x + 1)(x - 3)(x - 1)}. \] 4. **Cancel out common factors:** - The \((x - 1)\) in the numerator and denominator cancels out. - The \((x + 1)\) in the numerator and denominator cancels out. After canceling, we are left with: \[ \frac{-x(x - 1)}{(x - 3)(x - 3)} = \frac{-x(x - 1)}{(x - 3)^2}. \] Thus, the simplified form of the original expression is: \[ \frac{x - x^{2}}{x^{2} - 6x + 9}. \]