To solve this expression step by step, let’s rewrite and simplify it systematically. The given expression is:

$3xy^2 \cdot (-6) - 4x^2y \cdot (-4) + 4xy^2 \cdot \left(-\frac{1}{4}\right) + 4x^2y$

Step 1: Simplify each term

1. First term: $3xy^2 \cdot (-6)$ $= -18xy^2$

2. Second term: $-4x^2y \cdot (-4)$ $= 16x^2y$

3. Third term: $4xy^2 \cdot \left(-\frac{1}{4}\right)$ $= -xy^2$

4. Fourth term: $+4x^2y$ (No changes here; keep it as is.)

Step 2: Combine like terms

Now, group similar terms:

• Combine terms involving $xy^2$: $-18xy^2 - xy^2 = -19xy^2$
• Combine terms involving $x^2y$: $16x^2y + 4x^2y = 20x^2y$

Step 3: Write the simplified expression

The simplified form of the expression is: $-19xy^2 + 20x^2y$

This is the final simplified form.

Do you want details or have any questions?

Here are five related questions for deeper understanding:

1. What are like terms, and why are they grouped together during simplification?
2. How do the rules of multiplication and distribution apply in this problem?
3. How would the solution change if one of the coefficients was a fraction instead of an integer?
4. How can factoring be used to further analyze expressions like this?
5. How does the order of operations influence the simplification process?

Tip: Always distribute multiplication across terms carefully and check for negative signs to avoid errors during simplification!