1. **State the problem:** Simplify the expression $$\frac{x+8}{x^2+9x+20} + \frac{x+5}{x^2+7x+12}$$ ignoring restrictions on the variable. 2. **Factor the denominators:** $$x^2+9x+20 = (x+4)(x+5)$$ $$x^2+7x+12 = (x+3)(x+4)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{x+8}{(x+4)(x+5)} + \frac{x+5}{(x+3)(x+4)}$$ 4. **Find the common denominator:** The least common denominator (LCD) is $$(x+3)(x+4)(x+5)$$ 5. **Rewrite each fraction with the LCD:** $$\frac{(x+8)(x+3)}{(x+3)(x+4)(x+5)} + \frac{(x+5)(x+5)}{(x+3)(x+4)(x+5)}$$ 6. **Combine the numerators over the common denominator:** $$\frac{(x+8)(x+3) + (x+5)^2}{(x+3)(x+4)(x+5)}$$ 7. **Expand the numerators:** $$(x+8)(x+3) = x^2 + 3x + 8x + 24 = x^2 + 11x + 24$$ $$(x+5)^2 = x^2 + 10x + 25$$ 8. **Add the expanded numerators:** $$x^2 + 11x + 24 + x^2 + 10x + 25 = 2x^2 + 21x + 49$$ 9. **Rewrite the expression:** $$\frac{2x^2 + 21x + 49}{(x+3)(x+4)(x+5)}$$ 10. **Check if numerator factors:** Try to factor $2x^2 + 21x + 49$. Calculate discriminant: $$\Delta = 21^2 - 4 \times 2 \times 49 = 441 - 392 = 49$$ Since $\Delta$ is a perfect square, factorization is possible: $$2x^2 + 21x + 49 = (2x + 7)(x + 7)$$ 11. **Final simplified expression:** $$\frac{(2x + 7)(x + 7)}{(x+3)(x+4)(x+5)}$$ **Answer:** $$\boxed{\frac{(2x + 7)(x + 7)}{(x+3)(x+4)(x+5)}}$$