3x²yz - zx²y + 4yxz² - 2x²zy + 3z²yx - 3zy²

To simplify the given expression \(3x^2yz - zx^2y + 4yxz^2 - 2x^2zy + 3z^2yx - 3zy^2\), we can start by combining like terms. Let's rewrite the expression, grouping the terms with similar variables:

1. The terms \(3x^2yz\), \(-zx^2y\), and \(-2x^2zy\) all contain \(x^2yz\):
   \[
   3x^2yz - zx^2y - 2x^2zy = (3 - 1 - 2)x^2yz = 0
   \]

2. Now, we look at the remaining terms \(4yxz^2\) and \(3z^2yx\) which both have \(yxz^2\):
   \[
   4yxz^2 + 3z^2yx = (4 + 3)yxz^2 = 7yxz^2
   \]

3. Finally, we have \(-3zy^2\), which does not have a like term to combine with.

Now, we can rewrite the simplified expression with the results from each step:

\[
7yxz^2 - 3zy^2
\]

**Final simplified expression:**
\[
7yxz^2 - 3zy^2
\]

This is the simplified form of the given expression. Would you like further details or have any questions?

### Related Questions:

1. How do you factorize a polynomial expression with more than three variables?
2. What is the distributive property, and how is it used in polynomial simplification?
3. How can you identify like terms in a multivariable polynomial?
4. What are common techniques for simplifying complex algebraic expressions?
5. How can we verify the correctness of our simplification for multivariable expressions?

### Tip:
Always look for common factors or like terms when simplifying algebraic expressions to make the process easier.

Learn how to simplify the multivariable polynomial 3x²yz - zx²y + 4yxz² - 2x²zy + 3z²yx - 3zy² with step-by-step instructions. This problem involves polynomial simplification techniques and is suitable for high school and advanced students.

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