Find the least common denominator of \( \frac{2 x}{x^{2}+x-20} \) and \( \frac{3 x}{x^{2}+4 x-5} \).

To find the least common denominator (LCD) of the fractions \( \frac{2 x}{x^{2}+x-20} \) and \( \frac{3 x}{x^{2}+4 x-5} \), we first need to factor the denominators. 1. **Factor the first denominator**: \( x^{2}+x-20 \) We need to find two numbers that multiply to \(-20\) and add to \(1\). The numbers \(5\) and \(-4\) work because: \[ 5 \times (-4) = -20 \quad \text{and} \quad 5 + (-4) = 1 \] Thus, we can factor it as: \[ x^{2}+x-20 = (x+5)(x-4) \] 2. **Factor the second denominator**: \( x^{2}+4x-5 \) We need to find two numbers that multiply to \(-5\) and add to \(4\). The numbers \(5\) and \(-1\) work because: \[ 5 \times (-1) = -5 \quad \text{and} \quad 5 + (-1) = 4 \] Thus, we can factor it as: \[ x^{2}+4x-5 = (x+5)(x-1) \] 3. **Identify the unique factors**: The factored forms of the denominators are: - \( (x+5)(x-4) \) for \( x^{2}+x-20 \) - \( (x+5)(x-1) \) for \( x^{2}+4x-5 \) The unique factors from both denominators are: - \( (x+5) \) - \( (x-4) \) - \( (x-1) \) 4. **Construct the least common denominator**: The least common denominator (LCD) is the product of all unique factors: \[ \text{LCD} = (x+5)(x-4)(x-1) \] Thus, the least common denominator of \( \frac{2 x}{x^{2}+x-20} \) and \( \frac{3 x}{x^{2}+4 x-5} \) is: \[ \boxed{(x+5)(x-4)(x-1)} \]