1. **State the problem:** Simplify the expression $$\frac{7x + 3}{x - 4} + \frac{5x}{x^2 - 16}$$. 2. **Identify the denominators:** The denominators are $x - 4$ and $x^2 - 16$. Note that $x^2 - 16$ is a difference of squares and can be factored: $$x^2 - 16 = (x - 4)(x + 4)$$ 3. **Find the common denominator:** The least common denominator (LCD) is $(x - 4)(x + 4)$. 4. **Rewrite each fraction with the LCD:** - The first fraction needs to be multiplied by $\frac{x + 4}{x + 4}$: $$\frac{7x + 3}{x - 4} = \frac{(7x + 3)(x + 4)}{(x - 4)(x + 4)}$$ - The second fraction already has the denominator $(x - 4)(x + 4)$. 5. **Write the sum with the common denominator:** $$\frac{(7x + 3)(x + 4)}{(x - 4)(x + 4)} + \frac{5x}{(x - 4)(x + 4)} = \frac{(7x + 3)(x + 4) + 5x}{(x - 4)(x + 4)}$$ 6. **Expand the numerator:** $$ (7x + 3)(x + 4) = 7x \cdot x + 7x \cdot 4 + 3 \cdot x + 3 \cdot 4 = 7x^2 + 28x + 3x + 12 = 7x^2 + 31x + 12 $$ 7. **Add the remaining term in the numerator:** $$7x^2 + 31x + 12 + 5x = 7x^2 + 36x + 12$$ 8. **Write the full expression:** $$\frac{7x^2 + 36x + 12}{(x - 4)(x + 4)}$$ 9. **Factor the numerator if possible:** Find factors of $7 \times 12 = 84$ that sum to 36. Factors 6 and 14 sum to 20, 12 and 7 sum to 19, 21 and 4 sum to 25, 28 and 3 sum to 31, 42 and 2 sum to 44, 3 and 28 sum to 31, 6 and 14 sum to 20, 12 and 7 sum to 19, 21 and 4 sum to 25, 28 and 3 sum to 31, 42 and 2 sum to 44, 1 and 84 sum to 85. None sum to 36, so the quadratic does not factor nicely. 10. **Final simplified expression:** $$\frac{7x^2 + 36x + 12}{(x - 4)(x + 4)}$$ This is the simplified form of the original expression.