1. **State the problem:** Simplify the expression $$\frac{x + 6}{x^2 + 3x - 18} + \frac{1}{x - 3} + \frac{5}{2x + 12}$$. 2. **Factor denominators where possible:** - Factor the quadratic in the first denominator: $$x^2 + 3x - 18 = (x + 6)(x - 3)$$ - Factor the third denominator: $$2x + 12 = 2(x + 6)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{x + 6}{(x + 6)(x - 3)} + \frac{1}{x - 3} + \frac{5}{2(x + 6)}$$ 4. **Simplify the first fraction by canceling common factors:** $$\frac{\cancel{x + 6}}{\cancel{x + 6}(x - 3)} = \frac{1}{x - 3}$$ 5. **Rewrite the expression now:** $$\frac{1}{x - 3} + \frac{1}{x - 3} + \frac{5}{2(x + 6)}$$ 6. **Combine the first two fractions since they have the same denominator:** $$\frac{1}{x - 3} + \frac{1}{x - 3} = \frac{2}{x - 3}$$ 7. **Find the least common denominator (LCD) for the two remaining fractions:** - Denominators are $x - 3$ and $2(x + 6)$. - LCD is $2(x - 3)(x + 6)$. 8. **Rewrite each fraction with the LCD:** $$\frac{2}{x - 3} = \frac{2 \cdot 2(x + 6)}{2(x - 3)(x + 6)} = \frac{4(x + 6)}{2(x - 3)(x + 6)}$$ $$\frac{5}{2(x + 6)} = \frac{5(x - 3)}{2(x - 3)(x + 6)}$$ 9. **Add the fractions:** $$\frac{4(x + 6)}{2(x - 3)(x + 6)} + \frac{5(x - 3)}{2(x - 3)(x + 6)} = \frac{4(x + 6) + 5(x - 3)}{2(x - 3)(x + 6)}$$ 10. **Expand the numerator:** $$4(x + 6) + 5(x - 3) = 4x + 24 + 5x - 15 = 9x + 9$$ 11. **Factor the numerator:** $$9x + 9 = 9(x + 1)$$ 12. **Final simplified expression:** $$\frac{9(x + 1)}{2(x - 3)(x + 6)}$$ This is the simplified form of the original expression.