We need to factor the given expression:

\[
4a^2 b^2 - a^2 + 2ab - b^2
\]

### Step 1: Rearrange the terms
First, rewrite the expression to group terms involving \(a^2\), \(ab\), and \(b^2\):

\[
(4a^2 b^2 - a^2) + (2ab - b^2)
\]

### Step 2: Factor common terms
Let's factor out common terms from each group:

1. From \(4a^2 b^2 - a^2\), factor out \(a^2\):
   \[
   a^2(4b^2 - 1)
   \]
2. From \(2ab - b^2\), factor out \(b\):
   \[
   b(2a - b)
   \]

So, we now have:
\[
a^2(4b^2 - 1) + b(2a - b)
\]

### Step 3: Recognize special factor forms
Notice that \(4b^2 - 1\) is a difference of squares:
\[
4b^2 - 1 = (2b - 1)(2b + 1)
\]

Thus, the expression becomes:
\[
a^2(2b - 1)(2b + 1) + b(2a - b)
\]

At this point, the expression is factored as much as possible.

---

Would you like further clarification or details?

Here are five related questions to explore further:

1. How can the difference of squares be used in factoring?
2. Can grouping terms in different ways lead to different factorizations?
3. What is the next step if factoring by grouping doesn't work?
4. How do we apply factoring techniques to quadratic expressions?
5. How does recognizing patterns like the difference of squares simplify factoring?

**Tip:** When factoring, always check if a common factor can be factored out before using advanced techniques like difference of squares or grouping.

Learn how to factor the algebraic expression 4a^2 b^2 - a^2 + 2ab - b^2 using techniques like factoring by grouping and recognizing the difference of squares. A step-by-step guide suitable for students in Grades 9-10 covering key algebra concepts.

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