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You have provided the expression:

\[
\frac{4x^2y + 8x^2y^2 - xy^2}{4xy}
\]

Let's simplify this step-by-step.

### Step 1: Factor the numerator.
The given numerator is \( 4x^2y + 8x^2y^2 - xy^2 \). 

We can factor terms from each part:
- The first term \( 4x^2y \) has a factor of \( 4xy \),
- The second term \( 8x^2y^2 \) has a factor of \( 4xy \),
- The third term \( -xy^2 \) has a factor of \( -xy \).

Thus, the numerator can be rewritten as:
\[
4xy(x + 2xy - y).
\]

### Step 2: Rewrite the entire expression.
Now, the entire expression becomes:
\[
\frac{4xy(x + 2xy - y)}{4xy}.
\]

### Step 3: Cancel common factors.
The \( 4xy \) in the numerator and denominator cancel out, leaving us with:
\[
x + 2xy - y.
\]

Thus, the simplified expression is:

\[
x + 2xy - y.
\]

Would you like more details on the simplification steps or have any questions?

Here are 5 related questions to consider:
1. How would the expression change if an additional term is added to the numerator?
2. Can this expression be factored further if we treat it as a polynomial in terms of \(x\)?
3. How does the simplification process change if the denominator is something other than \(4xy\)?
4. What are the values of the simplified expression when \(x = 1\) and \(y = 2\)?
5. How can this expression be used in solving real-world algebraic problems?

**Tip:** Always look for common factors in both the numerator and denominator when simplifying fractions in algebra.

Simplify the expression: (4x^2y + 8x^2y^2 - xy^2) / 4xy.

Step-by-step guide on how to simplify the algebraic expression (4x^2y + 8x^2y^2 - xy^2) / 4xy by factoring and canceling common terms. This problem focuses on key algebraic concepts, including factoring and simplifying rational expressions. Suitable for Grades 9-10.

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