1. **State the problem:** Simplify the expression $$\frac{x^{2} + 14x + 49}{x^{2} + 15x + 56}$$. 2. **Recall the factoring formula:** For a quadratic expression $ax^2 + bx + c$, we look for two numbers that multiply to $c$ and add to $b$. 3. **Factor the numerator:** $x^{2} + 14x + 49$. - Find two numbers that multiply to 49 and add to 14: 7 and 7. - So, $x^{2} + 14x + 49 = (x + 7)(x + 7) = (x + 7)^2$. 4. **Factor the denominator:** $x^{2} + 15x + 56$. - Find two numbers that multiply to 56 and add to 15: 7 and 8. - So, $x^{2} + 15x + 56 = (x + 7)(x + 8)$. 5. **Rewrite the expression:** $$\frac{(x + 7)^2}{(x + 7)(x + 8)}$$. 6. **Simplify by canceling common factors:** Cancel $(x + 7)$ from numerator and denominator. - Result: $$\frac{x + 7}{x + 8}$$. 7. **Final answer:** The simplified form of the expression is $$\frac{x + 7}{x + 8}$$. This simplification is valid for all $x \neq -7$ and $x \neq -8$ to avoid division by zero.