https://file.aiquickdraw.com/mathbot/file/1725497125306i09k6yx0.jpg

The expression in the image is:

\[
3xy(2xy - 3x^3y - 4) - 6y^2x^4 - 2xy(9 - 2yx^2 - 8yx)
\]

Let's simplify this step by step.

### Step 1: Distribute Terms
1. **Expand \(3xy(2xy - 3x^3y - 4)\):**
   \[
   3xy \cdot 2xy = 6x^2y^2, \quad 3xy \cdot (-3x^3y) = -9x^4y^2, \quad 3xy \cdot (-4) = -12xy
   \]

   So, the expanded form of the first term is:
   \[
   6x^2y^2 - 9x^4y^2 - 12xy
   \]

2. **Expand \(-2xy(9 - 2yx^2 - 8yx)\):**
   \[
   -2xy \cdot 9 = -18xy, \quad -2xy \cdot (-2yx^2) = 4x^3y^2, \quad -2xy \cdot (-8yx) = 16x^2y^2
   \]

   So, the expanded form of the second term is:
   \[
   -18xy + 4x^3y^2 + 16x^2y^2
   \]

### Step 2: Combine Like Terms
Now, combine all the terms:
\[
6x^2y^2 - 9x^4y^2 - 12xy - 6x^4y^2 - 18xy + 4x^3y^2 + 16x^2y^2
\]

Combine like terms:
1. Combine \(x^2y^2\) terms: 
   \[
   6x^2y^2 + 16x^2y^2 = 22x^2y^2
   \]

2. Combine \(x^4y^2\) terms: 
   \[
   -9x^4y^2 + 4x^3y^2 - 6x^4y^2 = -6x^4y^2
   \]

3. Combine \(xy\) terms:
   \[
   -12xy - 18xy = -30xy
   \]

So, the simplified expression is:

\[
22x^2y^2 - 6x^4y^2 - 30xy
\]

Let me know if you would like further details or have any questions!

### Related Questions:
1. How do you factor polynomials with multiple variables?
2. What are the common techniques for expanding algebraic expressions?
3. How do you combine like terms involving exponents?
4. What is the role of distribution in simplifying expressions?
5. Can these terms be factored further?

### Tip:
Always group like terms carefully; it helps to rewrite similar terms next to each other to avoid mistakes when combining!

Simplify the expression: 3xy(2xy - 3x^3y - 4) - 6yx^4 - 2xy(9 - 2yx^2 - 8yx)

Learn how to simplify a complex algebraic expression by using the distributive property and combining like terms. This example covers essential algebraic concepts and is suitable for high school students in Grades 9-11.

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