Subtract the sum of `(9x-4y+6z^(2))` and `(-5x-4y-z^(2))` from the sum of `(6z+9y-2z^(2))` and `(-x-y-z^(2))`.

Options

A. `5x-16y-8z+6z^(2)`
B. `4x+14z^(2)-9y+7z`
C. `-5x+16y+6z-8z^(2)`
D. `5x-14y+8z-16z^(2)`

Answer

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

1. **Identify the expressions**: We have four expressions to work with:
- Expression A: \( 9x - 4y + 6z^2 \)
- Expression B: \( -5x - 4y - z^2 \)
- Expression C: \( 6z + 9y - 2z^2 \)
- Expression D: \( -x - y - z^2 \)

2. **Calculate the sum of Expression A and Expression B**:
\[ A + B = (9x - 4y + 6z^2) + (-5x - 4y - z^2) \]
Combine like terms:
\[ = (9x - 5x) + (-4y - 4y) + (6z^2 - z^2) \]
\[ = 4x - 8y + 5z^2 \]

3. **Calculate the sum of Expression C and Expression D**:
\[ C + D = (6z + 9y - 2z^2) + (-x - y - z^2) \]
Combine like terms:
\[ = (6z) + (9y - y) + (-2z^2 - z^2) + (-x) \]
\[ = -x + 6z + 8y - 3z^2 \]

4. **Subtract the sum of (A + B) from the sum of (C + D)**:
\[ (C + D) - (A + B) = (-x + 6z + 8y - 3z^2) - (4x - 8y + 5z^2) \]
Distribute the negative sign:
\[ = -x + 6z + 8y - 3z^2 - 4x + 8y - 5z^2 \]
Combine like terms:
\[ = (-x - 4x) + (6z) + (8y + 8y) + (-3z^2 - 5z^2) \]
\[ = -5x + 6z + 16y - 8z^2 \]

5. **Final Result**:
The final expression after performing the operations is:
\[ -5x + 6z + 16y - 8z^2 \]