The given expression is:

\[
x^2 - 10xy + 25y^2 - 1
\]

This resembles a quadratic expression. Let's analyze and simplify it step by step.

### Step 1: Recognize a Perfect Square
The first three terms, \(x^2 - 10xy + 25y^2\), form a perfect square trinomial:

\[
x^2 - 10xy + 25y^2 = (x - 5y)^2
\]

### Step 2: Rewrite the expression
Now, the expression becomes:

\[
(x - 5y)^2 - 1
\]

### Step 3: Apply the Difference of Squares formula
The expression is now in the form of a difference of squares:

\[
a^2 - b^2 = (a - b)(a + b)
\]

Here, \(a = (x - 5y)\) and \(b = 1\), so we can factor it as:

\[
(x - 5y - 1)(x - 5y + 1)
\]

### Final Factored Form:
The fully factored form of the expression is:

\[
(x - 5y - 1)(x - 5y + 1)
\]

Would you like a detailed explanation or have any further questions?

Here are five related questions to deepen your understanding:
1. How can the difference of squares method be used in other quadratic expressions?
2. How do you recognize a perfect square trinomial in algebraic expressions?
3. Can the difference of squares formula be applied in higher-degree polynomials?
4. What are some practical applications of factoring quadratic equations?
5. How would the factorization change if the original expression had a different constant term?

**Tip**: Always check for perfect square trinomials or difference of squares in quadratic expressions for easier factorization!

Learn how to factor the quadratic expression x² - 10xy + 25y² - 1 using algebraic techniques. This problem involves recognizing a perfect square trinomial and applying the difference of squares formula. Suitable for students in Grades 9-12.

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