Subtract the sum of `7x^(2) - 4y^(2) " and " 4x^(2) + 3xy - 4y^(2)` from the sum of `5x^(2) - 3xy - y^(2) "and " x^(2) + 2xy - 2y^(2)`.

Answer

To solve the problem, we need to follow these steps:

### Step 1: Calculate the sum of the first two expressions
We need to find the sum of \(7x^2 - 4y^2\) and \(4x^2 + 3xy - 4y^2\).

\[ (7x^2 - 4y^2) + (4x^2 + 3xy - 4y^2) \]

Combine like terms:
- Combine \(7x^2\) and \(4x^2\):
\[ 7x^2 + 4x^2 = 11x^2 \]
- Combine \(-4y^2\) and \(-4y^2\):
\[ -4y^2 - 4y^2 = -8y^2 \]
- The \(3xy\) term remains as it is.

Thus, the sum is:
\[ 11x^2 + 3xy - 8y^2 \]

### Step 2: Calculate the sum of the next two expressions
Now we need to find the sum of \(5x^2 - 3xy - y^2\) and \(x^2 + 2xy - 2y^2\).

\[ (5x^2 - 3xy - y^2) + (x^2 + 2xy - 2y^2) \]

Combine like terms:
- Combine \(5x^2\) and \(x^2\):
\[ 5x^2 + x^2 = 6x^2 \]
- Combine \(-3xy\) and \(2xy\):
\[ -3xy + 2xy = -xy \]
- Combine \(-y^2\) and \(-2y^2\):
\[ -y^2 - 2y^2 = -3y^2 \]

Thus, the sum is:
\[ 6x^2 - xy - 3y^2 \]

### Step 3: Subtract the first sum from the second sum
Now we need to subtract the first sum from the second sum:
\[ (6x^2 - xy - 3y^2) - (11x^2 + 3xy - 8y^2) \]

Distributing the negative sign:
\[ 6x^2 - xy - 3y^2 - 11x^2 - 3xy + 8y^2 \]

Combine like terms:
- Combine \(6x^2\) and \(-11x^2\):
\[ 6x^2 - 11x^2 = -5x^2 \]
- Combine \(-xy\) and \(-3xy\):
\[ -xy - 3xy = -4xy \]
- Combine \(-3y^2\) and \(8y^2\):
\[ -3y^2 + 8y^2 = 5y^2 \]

Thus, the final result is:
\[ -5x^2 - 4xy + 5y^2 \]

### Final Answer:
\[ -5x^2 - 4xy + 5y^2 \]

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