To simplify the given expression:

$\frac{1}{(x^2 - 1)^4} \left(\frac{6x^2 - 5}{1} - \frac{8x^3 - 20x + 4}{x^2 - 1}\right)$

Step 1: Simplify the expression inside the parentheses.

The expression inside the parentheses is:

$\frac{6x^2 - 5}{1} - \frac{8x^3 - 20x + 4}{x^2 - 1}$

This simplifies to:

$6x^2 - 5 - \frac{8x^3 - 20x + 4}{x^2 - 1}$

Step 2: Combine the terms by getting a common denominator.

The common denominator is $x^2 - 1$. So, rewrite the first term with the common denominator:

$\frac{(6x^2 - 5)(x^2 - 1)}{x^2 - 1} - \frac{8x^3 - 20x + 4}{x^2 - 1}$

This becomes:

$\frac{(6x^2 - 5)(x^2 - 1) - (8x^3 - 20x + 4)}{x^2 - 1}$

Step 3: Expand and simplify the numerator.

Expand $(6x^2 - 5)(x^2 - 1)$:

$(6x^2 - 5)(x^2 - 1) = 6x^4 - 6x^2 - 5x^2 + 5 = 6x^4 - 11x^2 + 5$

Now, substitute it back into the expression:

$\frac{6x^4 - 11x^2 + 5 - (8x^3 - 20x + 4)}{x^2 - 1}$

Simplify the numerator:

$6x^4 - 8x^3 - 11x^2 + 20x + 1$

Step 4: Substitute back into the original expression.

Now substitute this back into the original expression:

$\frac{1}{(x^2 - 1)^4} \cdot \frac{6x^4 - 8x^3 - 11x^2 + 20x + 1}{x^2 - 1}$

This simplifies to:

$\frac{6x^4 - 8x^3 - 11x^2 + 20x + 1}{(x^2 - 1)^5}$

So the simplified expression is:

$\frac{6x^4 - 8x^3 - 11x^2 + 20x + 1}{(x^2 - 1)^5}$

Would you like more details on any specific step, or do you have any further questions?

Here are some related questions you might consider:

1. How do you factor a higher degree polynomial like $6x^4 - 8x^3 - 11x^2 + 20x + 1$?
2. What are the properties of the function $\frac{1}{(x^2-1)^4}$ as $x$ approaches 1?
3. How can you find the roots of the polynomial in the numerator?
4. How do you find the derivative of a fraction with polynomials in both the numerator and the denominator?
5. How do you integrate a fraction of polynomials like the one obtained in the simplification?

Tip: Always look for common factors in both the numerator and the denominator when simplifying rational expressions.