To solve the inequality \(\left| x - 2 \right| \frac{x^2 - 1}{-(x^2 - 5x + 4)^3} > 0\), we need to analyze the sign of each factor individually and then determine where the entire expression is positive.

### Step 1: Analyze \(\left| x - 2 \right|\)
The absolute value function \(\left| x - 2 \right|\) is always non-negative, and it equals zero when \(x = 2\). Therefore, \(\left| x - 2 \right|\) is positive when \(x \neq 2\).

### Step 2: Analyze \(\frac{x^2 - 1}{-(x^2 - 5x + 4)^3}\)
This is the product of two factors: \(x^2 - 1\) and \(-(x^2 - 5x + 4)^3\).

#### Factor 1: \(x^2 - 1\)
- \(x^2 - 1\) can be factored as \((x - 1)(x + 1)\).
- \(x^2 - 1 > 0\) when \(x > 1\) or \(x < -1\).
- \(x^2 - 1 < 0\) when \(-1 < x < 1\).
- \(x^2 - 1 = 0\) when \(x = \pm 1\).

#### Factor 2: \(-(x^2 - 5x + 4)^3\)
- \(x^2 - 5x + 4\) can be factored as \((x - 4)(x - 1)\).
- \(x^2 - 5x + 4 = 0\) when \(x = 1\) or \(x = 4\).
- \((x^2 - 5x + 4)^3 > 0\) when \(x \in (1, 4)\).
- \((x^2 - 5x + 4)^3 < 0\) when \(x \in (-\infty, 1) \cup (4, \infty)\).
- The sign of the whole term \(-(x^2 - 5x + 4)^3\) will be opposite to the sign of \((x^2 - 5x + 4)^3\).

### Step 3: Determine where the entire expression is positive
The inequality is satisfied when the entire expression is positive. To determine this, we need to consider the intervals where the expression is positive.

- **When \(x \in (-\infty, -1)\)**:
  - \(\left| x - 2 \right| > 0\)
  - \(x^2 - 1 > 0\)
  - \(x^2 - 5x + 4 > 0\), so \(-(x^2 - 5x + 4)^3 < 0\)
  - Expression is **negative**.

- **When \(x \in (-1, 1)\)**:
  - \(\left| x - 2 \right| > 0\)
  - \(x^2 - 1 < 0\)
  - \(x^2 - 5x + 4 > 0\), so \(-(x^2 - 5x + 4)^3 < 0\)
  - Expression is **positive**.

- **When \(x \in (1, 2)\)**:
  - \(\left| x - 2 \right| > 0\)
  - \(x^2 - 1 < 0\)
  - \(x^2 - 5x + 4 < 0\), so \(-(x^2 - 5x + 4)^3 > 0\)
  - Expression is **negative**.

- **When \(x \in (2, 4)\)**:
  - \(\left| x - 2 \right| > 0\)
  - \(x^2 - 1 > 0\)
  - \(x^2 - 5x + 4 < 0\), so \(-(x^2 - 5x + 4)^3 > 0\)
  - Expression is **positive**.

- **When \(x \in (4, \infty)\)**:
  - \(\left| x - 2 \right| > 0\)
  - \(x^2 - 1 > 0\)
  - \(x^2 - 5x + 4 > 0\), so \(-(x^2 - 5x + 4)^3 < 0\)
  - Expression is **negative**.

### Step 4: Consider all cases and find the intervals
The expression is positive in the intervals where:
- \(x \in (-1, 1)\)
- \(x \in (2, 4)\)

### Conclusion
The solution to the inequality is:
\[
x \in (-1, 1) \cup (2, 4)
\]

Would you like further details or have any questions about this solution?

Here are some related questions you might find interesting:
1. How do you solve inequalities involving absolute values?
2. What are the general methods for solving polynomial inequalities?
3. Can you provide examples of inequalities that involve higher-degree polynomials?
4. How does the sign of a function's derivative relate to the function's intervals of increase and decrease?
5. How do we factor higher-degree polynomials effectively?

**Tip:** When solving inequalities, always consider the sign of each factor individually to understand how they contribute to the overall expression.

This detailed solution breaks down the steps to solve the polynomial inequality |x-2|(x^2-1) ÷ -(x^2-5x+4)^3 > 0. Learn how to factor polynomials, analyze sign changes, and apply absolute value properties. Suitable for high school students in Grades 10-12.

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