What should be added to product of `(x ^(2) + xy - y ^(2)) and (x ^(2) - xy + y ^(2)) ` to get `x ^(2) y ^(2)`

Answer

To solve the problem of what should be added to the product of \( (x^2 + xy - y^2) \) and \( (x^2 - xy + y^2) \) to get \( x^2y^2 \), we can follow these steps:

### Step 1: Calculate the product of the two expressions.

We need to multiply:
\[ (x^2 + xy - y^2)(x^2 - xy + y^2) \]

Using the distributive property (FOIL method), we get:
\[ = x^2(x^2) + x^2(-xy) + x^2(y^2) + xy(x^2) + xy(-xy) + xy(y^2) - y^2(x^2) - y^2(-xy) - y^2(y^2) \]

Calculating each term:
1. \( x^2 \cdot x^2 = x^4 \)
2. \( x^2 \cdot (-xy) = -x^3y \)
3. \( x^2 \cdot y^2 = x^2y^2 \)
4. \( xy \cdot x^2 = x^3y \)
5. \( xy \cdot (-xy) = -x^2y^2 \)
6. \( xy \cdot y^2 = xy^3 \)
7. \( -y^2 \cdot x^2 = -x^2y^2 \)
8. \( -y^2 \cdot (-xy) = xy^3 \)
9. \( -y^2 \cdot y^2 = -y^4 \)

Combining like terms:
\[ x^4 + (-x^3y + x^3y) + (x^2y^2 - x^2y^2 - x^2y^2) + (xy^3 + xy^3) - y^4 \]
This simplifies to:
\[ x^4 - x^2y^2 + 2xy^3 - y^4 \]

### Step 2: Set up the equation to find what should be added.

We want to find \( k \) such that:
\[ x^4 - x^2y^2 + 2xy^3 - y^4 + k = x^2y^2 \]

### Step 3: Rearrange the equation.

Rearranging gives:
\[ k = x^2y^2 - (x^4 - x^2y^2 + 2xy^3 - y^4) \]
\[ k = x^2y^2 - x^4 + x^2y^2 - 2xy^3 + y^4 \]
Combining like terms:
\[ k = 2x^2y^2 - x^4 - 2xy^3 + y^4 \]

### Step 4: Final answer.

Thus, the expression that should be added is:
\[ k = 2x^2y^2 - x^4 - 2xy^3 + y^4 \]