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

Answer

To solve the problem of adding the three polynomials \(2x^2 + 5xy - 2y^2\), \(3x^2 - 2xy + 5y^2\), and \(2x^2 + 8xy + 3y^2\), we will follow these steps:

### Step 1: Write down the polynomials
We start by writing down the three polynomials clearly:
1. \(2x^2 + 5xy - 2y^2\)
2. \(3x^2 - 2xy + 5y^2\)
3. \(2x^2 + 8xy + 3y^2\)

### Step 2: Group like terms
Next, we will group the like terms from all three polynomials. The like terms are the terms that contain the same variables raised to the same powers. We will group them as follows:
- \(x^2\) terms: \(2x^2\), \(3x^2\), and \(2x^2\)
- \(xy\) terms: \(5xy\), \(-2xy\), and \(8xy\)
- \(y^2\) terms: \(-2y^2\), \(5y^2\), and \(3y^2\)

### Step 3: Add the coefficients of like terms
Now, we will add the coefficients of each group:
1. For \(x^2\) terms:
\[ 2 + 3 + 2 = 7 \]
So, the total for \(x^2\) terms is \(7x^2\).

2. For \(xy\) terms:
\[ 5 - 2 + 8 = 11 \]
So, the total for \(xy\) terms is \(11xy\).

3. For \(y^2\) terms:
\[ -2 + 5 + 3 = 6 \]
So, the total for \(y^2\) terms is \(6y^2\).

### Step 4: Combine the results
Finally, we combine all the results from the previous step to form the final polynomial:
\[ 7x^2 + 11xy + 6y^2 \]

### Final Answer
The sum of the three polynomials is:
\[ \boxed{7x^2 + 11xy + 6y^2} \]