1. **State the problem:** Simplify the expression $$\frac{x}{x^2 + 5x + 6} + \frac{15}{x^2 + 9x + 14} - \frac{12}{x^2 + 10x + 21}$$ 2. **Factor each quadratic denominator:** - $x^2 + 5x + 6 = (x+2)(x+3)$ - $x^2 + 9x + 14 = (x+7)(x+2)$ - $x^2 + 10x + 21 = (x+7)(x+3)$ 3. **Rewrite the expression with factored denominators:** $$\frac{x}{(x+2)(x+3)} + \frac{15}{(x+7)(x+2)} - \frac{12}{(x+7)(x+3)}$$ 4. **Find the common denominator:** The least common denominator (LCD) is $(x+2)(x+3)(x+7)$. 5. **Rewrite each fraction with the LCD:** - First term: multiply numerator and denominator by $(x+7)$ $$\frac{x(x+7)}{(x+2)(x+3)(x+7)}$$ - Second term: multiply numerator and denominator by $(x+3)$ $$\frac{15(x+3)}{(x+7)(x+2)(x+3)}$$ - Third term: multiply numerator and denominator by $(x+2)$ $$\frac{12(x+2)}{(x+7)(x+3)(x+2)}$$ 6. **Combine the numerators over the common denominator:** $$\frac{x(x+7) + 15(x+3) - 12(x+2)}{(x+2)(x+3)(x+7)}$$ 7. **Expand the numerators:** - $x(x+7) = x^2 + 7x$ - $15(x+3) = 15x + 45$ - $12(x+2) = 12x + 24$ 8. **Combine like terms in the numerator:** $$x^2 + 7x + 15x + 45 - 12x - 24 = x^2 + (7x + 15x - 12x) + (45 - 24) = x^2 + 10x + 21$$ 9. **Rewrite the expression:** $$\frac{x^2 + 10x + 21}{(x+2)(x+3)(x+7)}$$ 10. **Factor the numerator:** $$x^2 + 10x + 21 = (x+7)(x+3)$$ 11. **Simplify the fraction by canceling common factors:** $$\frac{(x+7)(x+3)}{(x+2)(x+3)(x+7)} = \frac{1}{x+2}$$ **Final answer:** $$\boxed{\frac{1}{x+2}}$$