This expression is the sum of two rational functions:

$\frac{4}{x^2 - 16} + \frac{1}{x^2 + 5x + 4}.$

Step 1: Factorize the Denominators

1. $x^2 - 16$ is a difference of squares: $x^2 - 16 = (x - 4)(x + 4).$

2. $x^2 + 5x + 4$ is a quadratic trinomial: $x^2 + 5x + 4 = (x + 1)(x + 4).$

So, the expression becomes: $\frac{4}{(x - 4)(x + 4)} + \frac{1}{(x + 1)(x + 4)}.$

Step 2: Find a Common Denominator

The least common denominator (LCD) is: $(x - 4)(x + 4)(x + 1).$

Rewrite each fraction with this LCD: $\frac{4}{(x - 4)(x + 4)} = \frac{4(x + 1)}{(x - 4)(x + 4)(x + 1)},$ $\frac{1}{(x + 1)(x + 4)} = \frac{1(x - 4)}{(x - 4)(x + 4)(x + 1)}.$

Step 3: Combine the Fractions

Combine the two fractions under a single denominator: $\frac{4(x + 1)}{(x - 4)(x + 4)(x + 1)} + \frac{1(x - 4)}{(x - 4)(x + 4)(x + 1)} = \frac{4(x + 1) + (x - 4)}{(x - 4)(x + 4)(x + 1)}.$

Simplify the numerator: $4(x + 1) + (x - 4) = 4x + 4 + x - 4 = 5x.$

So, the result is: $\frac{5x}{(x - 4)(x + 4)(x + 1)}.$

Final Answer:

$\frac{5x}{(x - 4)(x + 4)(x + 1)}.$

Would you like further clarification or a detailed explanation of any step?

Related Questions:

1. How do you factorize more complex quadratics?
2. What are other examples of combining rational functions?
3. What happens if one denominator is already factored?
4. How does the least common denominator affect simplification?
5. How can we confirm the result by substitution?

Tip:

Always check for restrictions on $x$ (e.g., values that make the denominator zero). For this problem, $x \neq -4, -1, 4$.