How much is `-x^(2)+7y^(2)-3xy` less than `2x^(2)-y^(2)+xy`?

Options

A. `3x^(2)-8y^(2)+4xy`
B. `4x^(2)-8y^(2)+4xy`
C. `3x^(2)-5y^(2)+4xy`
D. `3x^(2)-8y^(2)-4xy`

Answer

To find out how much `-x^2 + 7y^2 - 3xy` is less than `2x^2 - y^2 + xy`, we need to calculate the difference between the two expressions.

### Step-by-step Solution:

1. **Identify the expressions**:
- Expression A: `2x^2 - y^2 + xy`
- Expression B: `-x^2 + 7y^2 - 3xy`

2. **Set up the subtraction**:
We want to find the difference:
\[ \text{Difference} = \text{Expression A} - \text{Expression B} \]

3. **Substitute the expressions**:
\[ \text{Difference} = (2x^2 - y^2 + xy) - (-x^2 + 7y^2 - 3xy) \]

4. **Distribute the negative sign**:
When we subtract Expression B, we change the signs of its terms:
\[ \text{Difference} = 2x^2 - y^2 + xy + x^2 - 7y^2 + 3xy \]

5. **Combine like terms**:
- Combine the `x^2` terms: \(2x^2 + x^2 = 3x^2\)
- Combine the `y^2` terms: \(-y^2 - 7y^2 = -8y^2\)
- Combine the `xy` terms: \(xy + 3xy = 4xy\)

Putting it all together:
\[ \text{Difference} = 3x^2 - 8y^2 + 4xy \]

6. **Final Answer**:
The expression `-x^2 + 7y^2 - 3xy` is less than `2x^2 - y^2 + xy` by:
\[ 3x^2 - 8y^2 + 4xy \]