To factor the expression \(x^3 - xy^2 + x^2 - y^3\), we'll try grouping the terms to find common factors.

First, let's rewrite the expression by grouping terms:

\[
(x^3 - y^3) + (x^2 - xy^2)
\]

Now, let's factor each group:

1. **Factor \(x^3 - y^3\)**: This is a difference of cubes, which factors as:
   \[
   x^3 - y^3 = (x - y)(x^2 + xy + y^2)
   \]

2. **Factor \(x^2 - xy^2\)**: We can factor out \(x^2\) from this term:
   \[
   x^2 - xy^2 = x^2(1 - y^2)
   \]
   Notice that \(1 - y^2\) is a difference of squares:
   \[
   1 - y^2 = (1 - y)(1 + y)
   \]
   So:
   \[
   x^2 - xy^2 = x^2(1 - y)(1 + y)
   \]

Now, let's combine the factored groups:

\[
x^3 - y^3 + x^2 - xy^2 = (x - y)(x^2 + xy + y^2) + x^2(1 - y)(1 + y)
\]

Notice that the expression still contains terms that are not factored completely. However, this is the simplest form we can achieve by grouping. This expression doesn't factor nicely into a single simple product.

So the factored form is:

\[
\boxed{(x - y)(x^2 + xy + y^2) + x^2(1 - y)(1 + y)}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you factor the expression \(x^3 + y^3\)?
2. What is the difference of cubes formula, and how is it applied?
3. How can you factor \(a^2 - b^2 + a - b\)?
4. What is the process for factoring a polynomial by grouping?
5. How would you factor the expression \(x^2y - y^3 + x^2 - xy\)?

**Tip:** Always check if the terms can be grouped in different ways to see if more simple factors can be obtained.

Learn how to factor the expression x^3 - xy^2 + x^2 - y^3 with a detailed step-by-step guide. This problem involves algebraic concepts such as polynomial factoring, difference of cubes, and difference of squares. Suitable for Grades 10-12.

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