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.