Find the least common denominator of \( \frac{-3}{x^{2}-4 x-21} \) and \( \frac{2}{x^{2}-4} \).

To find the least common denominator (LCD) of the two fractions, we first need to factor the denominators of each fraction: 1. **Factoring the first denominator \(x^{2}-4x-21\)**: We look for two numbers that multiply to \(-21\) (the constant term) and add up to \(-4\) (the coefficient of the \(x\) term). These numbers are \(-7\) and \(3\). Thus, we can factor the expression as follows: \[ x^{2}-4x-21 = (x-7)(x+3) \] 2. **Factoring the second denominator \(x^{2}-4\)**: This is a difference of squares, which can be factored as: \[ x^{2}-4 = (x-2)(x+2) \] Now we have: - The first denominator factors to \((x-7)(x+3)\) - The second denominator factors to \((x-2)(x+2)\) 3. **Finding the least common denominator**: The least common denominator (LCD) is found by taking each distinct factor from both denominators at its highest power. The factors we have are: - From the first denominator: \((x-7)\) and \((x+3)\) - From the second denominator: \((x-2)\) and \((x+2)\) Combining these factors, the least common denominator is: \[ \text{LCD} = (x-7)(x+3)(x-2)(x+2) \] Thus, the least common denominator of \( \frac{-3}{x^{2}-4 x-21} \) and \( \frac{2}{x^{2}-4} \) is: \[ \boxed{(x-7)(x+3)(x-2)(x+2)} \]