The given expression is:

\[
(x - y)^4 - 4(x - y)^2
\]

Let us simplify it step by step.

### Step 1: Use substitution
Let \( z = (x - y) \), so the expression becomes:

\[
z^4 - 4z^2
\]

### Step 2: Factorize the expression
This is a quadratic form in terms of \( z^2 \). We can factorize it as follows:

\[
z^4 - 4z^2 = z^2(z^2 - 4)
\]

### Step 3: Apply difference of squares
We notice that \( z^2 - 4 \) is a difference of squares:

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

### Step 4: Final factorization
Substitute back \( z = (x - y) \) to get:

\[
(x - y)^2 \left( (x - y) - 2 \right)\left( (x - y) + 2 \right)
\]

Thus, the fully factored form of the expression is:

\[
(x - y)^2 \left( (x - y) - 2 \right)\left( (x - y) + 2 \right)
\]

Would you like further details or have any questions about the steps?

Here are 5 related questions:
1. How can you factor more complex polynomial expressions?
2. What is the difference of squares rule and when is it useful?
3. How does substitution help in simplifying algebraic expressions?
4. Can the quadratic form be applied to other types of equations?
5. How do you expand factored expressions back into their original form?

**Tip:** Always check for common factor patterns such as difference of squares or perfect square trinomials when factoring.

This page provides a detailed step-by-step guide on how to factorize the expression (x - y)^4 - 4(x - y)^2. Key concepts include polynomial factorization and the difference of squares theorem. Suitable for students in Grades 10-12.

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