WARM-UP Find the least common denominator for the following rational expressions. \( \frac{18}{r-4} \cdot \frac{-5 r}{r^{2}-5 r+4} \cdot \frac{r+7}{r^{2}-4 r-5} \) \( \begin{array}{l}(r-1)(r-4)(r-5) \\ \mathbf{x}(r-1)(r+4)(r-5) \\ (r-1)(r+1)(r-4)(r-5) \\ (r-1)(r+1)(r+4)(r-5)\end{array} \)

We begin by factoring each denominator. 1. The first denominator is \[ r-4. \] 2. The second denominator is \[ r^2-5r+4. \] To factor this, we look for two numbers that multiply to \(4\) and add to \(-5\). The numbers \(-1\) and \(-4\) work since \[ -1 \times -4 = 4 \quad \text{and} \quad -1 + (-4) = -5. \] Thus, \[ r^2-5r+4 = (r-1)(r-4). \] 3. The third denominator is \[ r^2-4r-5. \] Here, we need two numbers that multiply to \(-5\) and add to \(-4\). The numbers \(-5\) and \(1\) satisfy these conditions since \[ -5 \times 1 = -5 \quad \text{and} \quad -5+1 = -4. \] Therefore, \[ r^2-4r-5 = (r-5)(r+1). \] Now, the list of distinct factors from all denominators is: - \(r-4\) (from the first and second expressions) - \(r-1\) (from the second expression) - \(r-5\) (from the third expression) - \(r+1\) (from the third expression) The least common denominator (LCD) is the product of all these factors: \[ (r-1)(r+1)(r-4)(r-5). \] Among the given choices, this corresponds to the third option: \[ (r-1)(r+1)(r-4)(r-5). \]