what must be subtracted from `3a^(2) - 6ab - 3b^(2) - 1` to get `4a^(2) - 7ab - 4b^(2) + 1`?

Answer

To solve the problem, we need to determine what must be subtracted from the expression \(3a^2 - 6ab - 3b^2 - 1\) to obtain the expression \(4a^2 - 7ab - 4b^2 + 1\).

### Step-by-Step Solution:

1. **Set Up the Equation**:
We want to find \(x\) such that:
\[ (3a^2 - 6ab - 3b^2 - 1) - x = (4a^2 - 7ab - 4b^2 + 1) \]

2. **Rearranging the Equation**:
To isolate \(x\), we can rearrange the equation:
\[ x = (3a^2 - 6ab - 3b^2 - 1) - (4a^2 - 7ab - 4b^2 + 1) \]

3. **Distributing the Negative Sign**:
Distributing the negative sign across the second expression:
\[ x = 3a^2 - 6ab - 3b^2 - 1 - 4a^2 + 7ab + 4b^2 - 1 \]

4. **Combining Like Terms**:
Now, we will combine the like terms:
- For \(a^2\) terms: \(3a^2 - 4a^2 = -1a^2\)
- For \(ab\) terms: \(-6ab + 7ab = 1ab\)
- For \(b^2\) terms: \(-3b^2 + 4b^2 = 1b^2\)
- For constant terms: \(-1 - 1 = -2\)

So, we have:
\[ x = -1a^2 + 1ab + 1b^2 - 2 \]

5. **Final Expression**:
We can rewrite the expression neatly:
\[ x = -a^2 + ab + b^2 - 2 \]

### Conclusion:
Thus, the expression that must be subtracted is:
\[ -a^2 + ab + b^2 - 2 \]