1. **State the problem:** Write $$\frac{3}{x+4} + \frac{5}{x^2 - 16}$$ as a single fraction in simplest form. 2. **Identify the denominators:** The denominators are $$x+4$$ and $$x^2 - 16$$. 3. **Factor the second denominator:** Note that $$x^2 - 16$$ is a difference of squares: $$x^2 - 16 = (x+4)(x-4)$$ 4. **Find the common denominator:** The least common denominator (LCD) is $$ (x+4)(x-4) $$. 5. **Rewrite each fraction with the LCD:** - The first fraction $$\frac{3}{x+4}$$ needs to be multiplied by $$\frac{x-4}{x-4}$$: $$\frac{3}{x+4} = \frac{3(x-4)}{(x+4)(x-4)}$$ - The second fraction already has the LCD: $$\frac{5}{(x+4)(x-4)}$$ 6. **Add the fractions:** $$\frac{3(x-4)}{(x+4)(x-4)} + \frac{5}{(x+4)(x-4)} = \frac{3(x-4) + 5}{(x+4)(x-4)}$$ 7. **Simplify the numerator:** $$3(x-4) + 5 = 3x - 12 + 5 = 3x - 7$$ 8. **Write the final simplified single fraction:** $$\frac{3x - 7}{(x+4)(x-4)}$$ **Answer:** $$\frac{3x - 7}{x^2 - 16}$$