1. **State the problem:** Simplify the expression $$\frac{x+3}{x^2 - 5x + 6} + \frac{6}{x^2 - 7x + 12}$$. 2. **Factor the denominators:** - Factor $$x^2 - 5x + 6$$ as $$(x-2)(x-3)$$. - Factor $$x^2 - 7x + 12$$ as $$(x-3)(x-4)$$. 3. **Rewrite the expression with factored denominators:** $$\frac{x+3}{(x-2)(x-3)} + \frac{6}{(x-3)(x-4)}$$. 4. **Find the common denominator:** The least common denominator (LCD) is $$(x-2)(x-3)(x-4)$$. 5. **Rewrite each fraction with the LCD:** - Multiply numerator and denominator of the first fraction by $$(x-4)$$: $$\frac{(x+3)(x-4)}{(x-2)(x-3)(x-4)}$$. - Multiply numerator and denominator of the second fraction by $$(x-2)$$: $$\frac{6(x-2)}{(x-3)(x-4)(x-2)}$$. 6. **Combine the fractions:** $$\frac{(x+3)(x-4) + 6(x-2)}{(x-2)(x-3)(x-4)}$$. 7. **Expand the numerators:** - $$(x+3)(x-4) = x^2 - 4x + 3x - 12 = x^2 - x - 12$$. - $$6(x-2) = 6x - 12$$. 8. **Add the numerators:** $$x^2 - x - 12 + 6x - 12 = x^2 + 5x - 24$$. 9. **Rewrite the expression:** $$\frac{x^2 + 5x - 24}{(x-2)(x-3)(x-4)}$$. 10. **Factor the numerator:** $$x^2 + 5x - 24 = (x+8)(x-3)$$. 11. **Simplify by canceling common factors:** Cancel $$(x-3)$$ from numerator and denominator: $$\frac{(x+8)\cancel{(x-3)}}{(x-2)\cancel{(x-3)}(x-4)} = \frac{x+8}{(x-2)(x-4)}$$. **Final answer:** $$\boxed{\frac{x+8}{(x-2)(x-4)}}$$