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To solve this problem, let's find the quotient by dividing the two rational expressions given.

The problem is:

\[
\frac{x^2 - 4}{x^2 + 8x + 16} \div \frac{4x - 8}{x^2 + 5x + 4}
\]

### Step 1: Rewrite as Multiplication
To divide by a fraction, we multiply by its reciprocal:

\[
\frac{x^2 - 4}{x^2 + 8x + 16} \times \frac{x^2 + 5x + 4}{4x - 8}
\]

### Step 2: Factor Each Expression
1. **Numerator of the first fraction:** \( x^2 - 4 = (x + 2)(x - 2) \) (difference of squares).
2. **Denominator of the first fraction:** \( x^2 + 8x + 16 = (x + 4)^2 \) (perfect square trinomial).
3. **Numerator of the second fraction:** \( x^2 + 5x + 4 = (x + 4)(x + 1) \) (factoring the quadratic).
4. **Denominator of the second fraction:** \( 4x - 8 = 4(x - 2) \).

Substituting the factored forms, we get:

\[
\frac{(x + 2)(x - 2)}{(x + 4)^2} \times \frac{(x + 4)(x + 1)}{4(x - 2)}
\]

### Step 3: Cancel Common Factors
Now we can cancel any terms that appear in both the numerator and denominator:
- \( (x - 2) \) cancels out from both a numerator and a denominator.
- \( (x + 4) \) in the denominator of the first fraction cancels with one \( (x + 4) \) in the numerator of the second fraction.

This simplifies to:

\[
\frac{(x + 2)(x + 1)}{4(x + 4)}
\]

### Final Answer
The simplified quotient is:

\[
\frac{(x + 2)(x + 1)}{4(x + 4)}
\]

Looking at the answer choices, this matches the second option. 

Would you like further clarification on any part of the solution?

### Related Questions
1. How do you factor quadratics with no common factors?
2. What is the difference between multiplying and dividing rational expressions?
3. How do you identify common factors in rational expressions?
4. What is the importance of factoring in simplifying rational expressions?
5. How do you handle cases where terms don’t cancel out in rational expressions?

### Tip
Always factor expressions completely before attempting to simplify or cancel terms. This can save time and make the solution clearer.

Find the quotient of (x^2 - 4) / (x^2 + 8x + 16) divided by (4x - 8) / (x^2 + 5x + 4).

Learn how to find the quotient of two rational expressions: (x^2 - 4) / (x^2 + 8x + 16) and (4x - 8) / (x^2 + 5x + 4). This problem involves factoring, simplifying expressions, and dividing rational functions. Suitable for Grade 10 students.

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