Let's simplify the expression \( \frac{5}{x^2 - 16} + \frac{2x}{x + 4} \).

### Step 1: Factor the denominator of the first fraction
The denominator \( x^2 - 16 \) is a difference of squares and can be factored as follows:
\[
x^2 - 16 = (x - 4)(x + 4)
\]
So, the expression becomes:
\[
\frac{5}{(x - 4)(x + 4)} + \frac{2x}{x + 4}
\]

### Step 2: Find a common denominator
The common denominator is \( (x - 4)(x + 4) \).

Now, rewrite the second fraction with the common denominator:
\[
\frac{2x}{x + 4} = \frac{2x(x - 4)}{(x + 4)(x - 4)}
\]

### Step 3: Combine the fractions
Now, we can add the fractions since they have a common denominator:
\[
\frac{5}{(x - 4)(x + 4)} + \frac{2x(x - 4)}{(x - 4)(x + 4)} = \frac{5 + 2x(x - 4)}{(x - 4)(x + 4)}
\]

### Step 4: Simplify the numerator
Expand and simplify the numerator:
\[
2x(x - 4) = 2x^2 - 8x
\]
So, the numerator becomes:
\[
5 + 2x^2 - 8x
\]
Thus, the expression simplifies to:
\[
\frac{2x^2 - 8x + 5}{(x - 4)(x + 4)}
\]

This is the simplified form of the expression \( \frac{5}{x^2 - 16} + \frac{2x}{x + 4} \).

Would you like more details or have any questions?

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Here are 5 related questions to deepen your understanding:

1. How would you simplify \( \frac{3x^2 + 7x - 2}{x^2 - 4} + \frac{2x + 1}{x + 2} \)?
2. Can you factor the expression \( x^4 - 16 \)? If so, how?
3. What is the result when simplifying \( \frac{x^2 - 9}{x^2 - 25} + \frac{5x}{x - 5} \)?
4. How do you find the domain of the function \( f(x) = \frac{5}{x^2 - 16} + \frac{2x}{x + 4} \)?
5. What happens if \( x = 4 \) is substituted into the original expression?

**Tip:** When simplifying algebraic fractions, always factor denominators and numerators where possible to identify common terms.

Learn how to simplify the expression 5/(x^2 - 16) + 2x/(x + 4) step-by-step. This problem involves algebraic concepts such as factoring, finding common denominators, and simplifying fractional expressions. Suitable for Grades 9-12 students.

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