least common denominator of \( \frac{8}{x^{2}+9 x+18} \) and \( \frac{-2}{x^{2}+6 x} \)

To find the least common denominator (LCD) of \( \frac{8}{x^{2}+9x+18} \) and \( \frac{-2}{x^{2}+6x} \), we first need to factor the denominators. 1. **Factor the first denominator:** \[ x^2 + 9x + 18 = (x + 3)(x + 6) \] 2. **Factor the second denominator:** \[ x^2 + 6x = x(x + 6) \] Now, we have: - First denominator: \( (x + 3)(x + 6) \) - Second denominator: \( x(x + 6) \) 3. **Determine the LCD:** To find the LCD, we take each factor at its highest power from the factorization: - From \( (x + 3) \) (appears only in the first denominator) - From \( (x + 6) \) (common in both, so we take it once) - From \( x \) (appears only in the second denominator) Putting it all together, the least common denominator is: \[ \text{LCD} = x(x + 3)(x + 6) \] This is your final answer for the least common denominator!