Step 1: Write the Expressions
Before performing the subtraction, let's express the terms clearly:
- First Expression:
\( 2x(x + y + z) - 2y(x - y - z) \)
- Second Expression:
\( 3z(-x + y + z) \)
Step 2: Expand the Expressions
Now we need to expand both expressions.
- Expanding the First Expression:
- \( 2x(x + y + z) = 2x^2 + 2xy + 2xz \)
- \( -2y(x - y - z) = -2yx + 2y^2 + 2yz \)
- Therefore,
\( 2x(x + y + z) - 2y(x - y - z) = 2x^2 + 2xy + 2xz - 2yx + 2y^2 + 2yz = 2x^2 + 2y^2 + 2xz + 2yz \)
- Expanding the Second Expression:
- \( 3z(-x + y + z) = -3zx + 3zy + 3z^2 \)
Step 3: Perform the Subtraction
Now, we subtract the first expression from the second:
- Subtraction:
\( 3z(-x + y + z) - (2x(x + y + z) - 2y(x - y - z)) \)
\( = -3zx + 3zy + 3z^2 - (2x^2 + 2y^2 + 2xz + 2yz) \)
Combine like terms:
- \( = -3zx + 3zy + 3z^2 - 2x^2 - 2y^2 - 2xz - 2yz \)
- \( = -2x^2 - 2y^2 - (3zx + 2xz) + (3zy - 2yz) + 3z^2 \)
Final Result
The final expression after simplification is:
- Final Expression:
\( -2x^2 - 2y^2 - 5zx + zy + 3z^2 \)
This is the result after subtracting the first expression from the second.