The sum of ` 6xy ( 2 x - 4z) , 3yz ( 2 x - 3 z)` and ` 4 xz (3 y - 2 y^(2))` is

Options

A. `12 x^(2)y - 6xyz - 9yz^(2) + 8x y^(2) z `
B. ` 12 x^(2) y - 6xyz + 9yz^(2) - 8xy^(2) z`
C. `12 x^(2) y - 6 xyz - 9yz^(2) - 8 x y^(2) z`
D. `12 x^(2) y + 6 xyz - 9yz^(2) - 8 x y^(2) z`

Answer

To find the sum of the expressions \( 6xy(2x - 4z) \), \( 3yz(2x - 3z) \), and \( 4xz(3y - 2y^2) \), we will follow these steps:

### Step 1: Expand each expression
1. **Expand \( 6xy(2x - 4z) \)**:
\[ 6xy(2x) - 6xy(4z) = 12x^2y - 24xyz \]

2. **Expand \( 3yz(2x - 3z) \)**:
\[ 3yz(2x) - 3yz(3z) = 6xyz - 9yz^2 \]

3. **Expand \( 4xz(3y - 2y^2) \)**:
\[ 4xz(3y) - 4xz(2y^2) = 12xyz - 8xy^2z \]

### Step 2: Combine all the expanded expressions
Now, we will combine all the results from the expansions:
\[ (12x^2y - 24xyz) + (6xyz - 9yz^2) + (12xyz - 8xy^2z) \]

### Step 3: Combine like terms
1. **Combine the \( xyz \) terms**:
\[ -24xyz + 6xyz + 12xyz = -24xyz + 18xyz = -6xyz \]

2. **Write down the other terms**:
\[ 12x^2y - 9yz^2 - 8xy^2z \]

### Step 4: Write the final expression
Putting it all together, we have:
\[ 12x^2y - 6xyz - 9yz^2 - 8xy^2z \]

### Final Result
The sum of the expressions is:
\[ 12x^2y - 6xyz - 9yz^2 - 8xy^2z \]

---