1. **State the problem:** We need to rewrite the sum of two algebraic fractions with different denominators by finding a common denominator and expressing both fractions with that denominator. 2. **Given expressions:** $$\frac{(2x - 3)(x + 1)}{x(x - 1)(x - 1)} + \frac{(3x + 2)(x - 1)}{x(x - 1)(x + 1)}$$ 3. **Identify the denominators:** - First denominator: $x(x - 1)(x - 1) = x(x - 1)^2$ - Second denominator: $x(x - 1)(x + 1)$ 4. **Find the least common denominator (LCD):** The LCD must include each factor the greatest number of times it appears in any denominator. - $x$ appears once in both. - $(x - 1)$ appears twice in the first denominator and once in the second, so take $(x - 1)^2$. - $(x + 1)$ appears once in the second denominator. Therefore, the LCD is: $$x(x - 1)^2(x + 1)$$ 5. **Rewrite each fraction with the LCD:** - First fraction already has denominator $x(x - 1)^2$, so multiply numerator and denominator by $(x + 1)$: $$\frac{(2x - 3)(x + 1) \times (x + 1)}{x(x - 1)^2 (x + 1)} = \frac{(2x - 3)(x + 1)^2}{x(x - 1)^2 (x + 1)}$$ - Second fraction has denominator $x(x - 1)(x + 1)$, missing one $(x - 1)$ to match $(x - 1)^2$, so multiply numerator and denominator by $(x - 1)$: $$\frac{(3x + 2)(x - 1) \times (x - 1)}{x(x - 1)(x + 1)(x - 1)} = \frac{(3x + 2)(x - 1)^2}{x(x - 1)^2 (x + 1)}$$ 6. **Final rewritten expression:** $$\frac{(2x - 3)(x + 1)^2}{x(x - 1)^2 (x + 1)} + \frac{(3x + 2)(x - 1)^2}{x(x - 1)^2 (x + 1)}$$ This matches the option: $$\frac{(2x - 3)(x + 1)}{x(x - 1)(x + 1)} + \frac{(3x + 2)(x - 1)}{x(x + 1)(x - 1)}$$ which is the second option in the list. **Answer:** The correct rewritten expression with the common denominator is: $$\frac{(2x - 3)(x + 1)}{x(x - 1)(x + 1)} + \frac{(3x + 2)(x - 1)}{x(x + 1)(x - 1)}$$