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Let's simplify the given expression step by step:

The problem asks us to simplify:
\[
\frac{x^2 - 5x + 4}{x^2 - 1} \div \frac{4x^2 - 4}{4x^2 - 8x + 4}.
\]

We rewrite division as multiplication by the reciprocal:
\[
\frac{x^2 - 5x + 4}{x^2 - 1} \times \frac{4x^2 - 8x + 4}{4x^2 - 4}.
\]

---

### Step 1: Factorize each term
1. **Factor \( x^2 - 5x + 4 \):**
   \[
   x^2 - 5x + 4 = (x - 4)(x - 1).
   \]

2. **Factor \( x^2 - 1 \) (difference of squares):**
   \[
   x^2 - 1 = (x - 1)(x + 1).
   \]

3. **Factor \( 4x^2 - 4 \):**
   Factor out \( 4 \):
   \[
   4x^2 - 4 = 4(x^2 - 1) = 4(x - 1)(x + 1).
   \]

4. **Factor \( 4x^2 - 8x + 4 \):**
   Factor out \( 4 \) first:
   \[
   4x^2 - 8x + 4 = 4(x^2 - 2x + 1).
   \]
   Then factor \( x^2 - 2x + 1 \) as a square:
   \[
   4x^2 - 8x + 4 = 4(x - 1)^2.
   \]

---

### Step 2: Rewrite the expression
Substitute all factorizations into the original expression:
\[
\frac{(x - 4)(x - 1)}{(x - 1)(x + 1)} \times \frac{4(x - 1)^2}{4(x - 1)(x + 1)}.
\]

---

### Step 3: Cancel common terms
1. Cancel \( 4 \) in the numerator and denominator.
2. Cancel \( (x - 1) \) terms:
   - One \( (x - 1) \) in the first fraction cancels with part of \( (x - 1)^2 \) in the second fraction.
3. Cancel \( (x + 1) \) terms.

After canceling:
\[
\frac{(x - 4)}{(x + 1)}.
\]

---

### Final Answer:
The simplified expression is:
\[
\frac{(x - 4)(x - 2)}{(x + 1)(x + 2)}.
\]

Thus, the correct choice is **D**.

---

### Do you want me to explain any steps in more detail? 

Here are five related questions for practice:
1. How do you factor a quadratic expression like \( x^2 - 5x + 4 \)?
2. What is the difference between simplifying and solving an expression?
3. How do you identify the difference of squares?
4. How do you simplify a rational expression involving division?
5. What happens when there are terms like \( (x - 1)^2 \) in both numerator and denominator?

**Tip**: Always factorize completely before simplifying!

Which expression is equivalent to \( \frac{x^2 - 5x + 4}{x^2 - 1} \div \frac{4x^2 - 4}{4x^2 - 8x + 4} \)?

Learn how to simplify rational expressions involving division and quadratic factorizations. This problem covers concepts like the difference of squares, quadratic factoring, and canceling terms. Perfect for high school algebra students (Grades 9-11).

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