What must be added to `2x^(2)-3xy+5y^(2)` to get `x^(2)-xy-y^(2)`?

Answer

To find out what must be added to the expression \(2x^2 - 3xy + 5y^2\) to obtain the expression \(x^2 - xy - y^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Set Up the Equation**: We need to find an expression \(A\) such that:
\[ 2x^2 - 3xy + 5y^2 + A = x^2 - xy - y^2 \]
To find \(A\), we can rearrange this equation:
\[ A = (x^2 - xy - y^2) - (2x^2 - 3xy + 5y^2) \]

2. **Distribute the Negative Sign**: We will distribute the negative sign across the second expression:
\[ A = x^2 - xy - y^2 - 2x^2 + 3xy - 5y^2 \]

3. **Combine Like Terms**: Now, we will combine the like terms:
- For \(x^2\) terms: \(x^2 - 2x^2 = -x^2\)
- For \(xy\) terms: \(-xy + 3xy = 2xy\)
- For \(y^2\) terms: \(-y^2 - 5y^2 = -6y^2\)

Putting it all together:
\[ A = -x^2 + 2xy - 6y^2 \]

4. **Final Answer**: Thus, the expression that must be added is:
\[ -x^2 + 2xy - 6y^2 \]