Simplify combining like terms:
`5x^(2)y - 5x^(2) + 3yx^(2) - 3y^(2) + x^(2) - y^(2) + 8xy^(2) - 3y^(2)`

Answer

To simplify the expression \( 5x^2y - 5x^2 + 3yx^2 - 3y^2 + x^2 - y^2 + 8xy^2 - 3y^2 \), we will combine like terms step by step.

### Step 1: Rewrite the expression
First, let's rewrite the expression for clarity:
\[ 5x^2y - 5x^2 + 3yx^2 - 3y^2 + x^2 - y^2 + 8xy^2 - 3y^2 \]

### Step 2: Identify and combine like terms
Now, we will group the like terms together.

1. **Like terms in \(x^2y\)**:
- \(5x^2y\) and \(3yx^2\) (which is the same as \(3x^2y\)):
\[ 5x^2y + 3x^2y = 8x^2y \]

2. **Like terms in \(x^2\)**:
- \(-5x^2\) and \(x^2\):
\[ -5x^2 + x^2 = -4x^2 \]

3. **Like terms in \(y^2\)**:
- \(-3y^2\), \(-y^2\), and \(-3y^2\):
\[ -3y^2 - y^2 - 3y^2 = -7y^2 \]

4. **The term with \(xy^2\)**:
- \(8xy^2\) remains as it is since there are no other \(xy^2\) terms to combine.

### Step 3: Combine all the results
Now we can combine all the simplified terms together:
\[ 8x^2y - 4x^2 - 7y^2 + 8xy^2 \]

### Final Answer
Thus, the simplified expression is:
\[ 8x^2y - 4x^2 - 7y^2 + 8xy^2 \]

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