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The expression provided is:

\[
\frac{w^8 - w^4 - 12}{w^8 + w^4 - 20}
\]

To simplify this expression, we'll start by factoring both the numerator and the denominator.

### 1. Factoring the numerator:
The numerator is \( w^8 - w^4 - 12 \). Let's set \( x = w^4 \) to simplify the factoring process:

\[
x^2 - x - 12
\]

Now, factor this quadratic expression:

\[
x^2 - x - 12 = (x - 4)(x + 3)
\]

Substituting back \( x = w^4 \):

\[
w^8 - w^4 - 12 = (w^4 - 4)(w^4 + 3)
\]

Notice that \( w^4 - 4 \) can be further factored as a difference of squares:

\[
w^4 - 4 = (w^2 - 2)(w^2 + 2)
\]

So the complete factorization of the numerator is:

\[
w^8 - w^4 - 12 = (w^2 - 2)(w^2 + 2)(w^4 + 3)
\]

### 2. Factoring the denominator:
The denominator is \( w^8 + w^4 - 20 \). Again, substitute \( x = w^4 \):

\[
x^2 + x - 20
\]

Factor this quadratic expression:

\[
x^2 + x - 20 = (x - 4)(x + 5)
\]

Substituting back \( x = w^4 \):

\[
w^8 + w^4 - 20 = (w^4 - 4)(w^4 + 5)
\]

Again, \( w^4 - 4 \) can be factored as:

\[
w^4 - 4 = (w^2 - 2)(w^2 + 2)
\]

So the complete factorization of the denominator is:

\[
w^8 + w^4 - 20 = (w^2 - 2)(w^2 + 2)(w^4 + 5)
\]

### 3. Simplifying the expression:
The expression now becomes:

\[
\frac{(w^2 - 2)(w^2 + 2)(w^4 + 3)}{(w^2 - 2)(w^2 + 2)(w^4 + 5)}
\]

Here, \( (w^2 - 2) \) and \( (w^2 + 2) \) cancel out, leaving:

\[
\frac{w^4 + 3}{w^4 + 5}
\]

This is the simplified form of the given expression.

Would you like more details, or do you have any questions?

### Relative Questions:
1. How do you factor quadratic expressions?
2. What are the steps to factor a difference of squares?
3. How can substitution simplify the factoring of polynomials?
4. Why does canceling common factors work in rational expressions?
5. How can you verify the simplification of a rational expression?

### Tip:
Always look for opportunities to factor out common terms when simplifying rational expressions—it can greatly simplify your work.

Learn how to simplify the rational expression (w^8 - w^4 - 12) / (w^8 + w^4 - 20) using factoring techniques. This problem involves factoring quadratic expressions and canceling common factors in rational expressions. Suitable for high school students studying algebra and polynomial expressions.

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