Add : `5x^(2)-2xy +8y^(2), 3xy -7y^(2)-2x^(2) and y^(2)+xy-4x^(2)`.

Answer

To solve the problem of adding the algebraic expressions \(5x^{2} - 2xy + 8y^{2}\), \(3xy - 7y^{2} - 2x^{2}\), and \(y^{2} + xy - 4x^{2}\), we will follow these steps:

### Step 1: Write down the expressions
We have the following three expressions to add:
1. \(5x^{2} - 2xy + 8y^{2}\)
2. \(3xy - 7y^{2} - 2x^{2}\)
3. \(y^{2} + xy - 4x^{2}\)

### Step 2: Combine like terms
We will combine all the like terms from the three expressions.

#### Grouping the terms:
- **\(x^{2}\) terms**: \(5x^{2} - 2x^{2} - 4x^{2}\)
- **\(xy\) terms**: \(-2xy + 3xy + xy\)
- **\(y^{2}\) terms**: \(8y^{2} - 7y^{2} + y^{2}\)

### Step 3: Calculate each group
1. **For \(x^{2}\) terms**:
\[ 5x^{2} - 2x^{2} - 4x^{2} = (5 - 2 - 4)x^{2} = -1x^{2} = -x^{2} \]

2. **For \(xy\) terms**:
\[ -2xy + 3xy + xy = (-2 + 3 + 1)xy = 2xy \]

3. **For \(y^{2}\) terms**:
\[ 8y^{2} - 7y^{2} + y^{2} = (8 - 7 + 1)y^{2} = 2y^{2} \]

### Step 4: Combine the results
Now we combine all the results from the calculations:
\[ -x^{2} + 2xy + 2y^{2} \]

### Final Answer
Thus, the sum of the three expressions is:
\[ -x^{2} + 2xy + 2y^{2} \]

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