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Understanding algebraic expressions is crucial for solving various types of mathematical problems. An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operations (such as addition, subtraction, multiplication, and division). The expression represents a value that can change depending on the variables involved.

For instance, in the exercise \( (x-y)^4 - 4(x-y)^2 \) the terms \( (x-y)^4 \) and \( -4(x-y)^2 \) are algebraic expressions that involve a variable part \( (x-y) \) raised to a power. Identifying the structure of algebraic expressions is the first step towards factoring. This includes recognizing the form \( a^2 - b^2 \) which signifies a difference of squares, a pivotal concept in algebra that simplifies factoring complex expressions.

Polynomial factoring is a process of breaking down a polynomial expression into simpler components that, when multiplied together, give you the original polynomial. This is an essential skill in algebra as it can simplify expressions and solve equations.

When it comes to factoring, there are different forms and strategies, such as pulling out the greatest common factor, grouping, or using special products, including the difference of squares. Factoring the given exercise \( (x-y)^4 - 4(x-y)^2 \) starts with recognizing patterns that match these strategies. Recognizing that \( (x-y)^4 \) is a square of \( (x-y)^2 \) and 4 is a square of 2, helps us employ the difference of squares technique to factor the polynomial effectively.

The difference of squares formula is one of the most widely used factoring techniques in algebra. It states that if you have two terms and each is a perfect square, you can factor them as the product of their sum and difference: \( a^2 - b^2 = (a - b)(a + b) \).

In our exercise, we identified \( a = (x-y)^2 \) and \( b = 2 \) to fit this formula. After applying it, we simplified the expression to \( ((x-y)^2 - 2)((x-y)^2 + 2) \) as Step 2 suggests, and further factored \( (x-y)^2 - 2 \) using the sum and difference of two squares individually. This completes the process, providing us with the fully factored form of the original polynomial.