What should be subtracted from `2x^(2)-y^(2)+4z^(2)` to get `x^(2)+y^(2)-z^(2)`?

Answer

To solve the problem, we need to determine what should be subtracted from the expression \(2x^2 - y^2 + 4z^2\) to yield the expression \(x^2 + y^2 - z^2\).

### Step-by-Step Solution:

1. **Set up the equation**:
We want to find an expression \(A\) such that:
\[ (2x^2 - y^2 + 4z^2) - A = x^2 + y^2 - z^2 \]

2. **Rearranging the equation**:
To find \(A\), we can rearrange the equation:
\[ A = (2x^2 - y^2 + 4z^2) - (x^2 + y^2 - z^2) \]

3. **Distributing the negative sign**:
Now, we will distribute the negative sign across the second expression:
\[ A = 2x^2 - y^2 + 4z^2 - x^2 - y^2 + z^2 \]

4. **Combining like terms**:
Now, we combine the like terms:
- For \(x^2\): \(2x^2 - x^2 = x^2\)
- For \(y^2\): \(-y^2 - y^2 = -2y^2\)
- For \(z^2\): \(4z^2 + z^2 = 5z^2\)

Therefore, we have:
\[ A = x^2 - 2y^2 + 5z^2 \]

5. **Final Answer**:
The expression that should be subtracted from \(2x^2 - y^2 + 4z^2\) to get \(x^2 + y^2 - z^2\) is:
\[ x^2 - 2y^2 + 5z^2 \]