Solve the expression `(2x + 3y - 4z ) ^(2).`

Options

A. `4x ^(2) + 9y ^(2) + 16 z ^(2) + 12 xy`
B. `4x ^(2) + 9y ^(2) + 16 z ^(2) + 12 xy + 23 yz - 16 zx`
C. `4x ^(2) + 9y ^(2) + 16 z ^(2) + 12 xy - 24 yz + 16 zx`
D. None of the above

Answer

To solve the expression \((2x + 3y - 4z)^2\), we will use the formula for the square of a trinomial, which is given by:

\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \]

In our case, we can identify:
- \(a = 2x\)
- \(b = 3y\)
- \(c = -4z\)

Now, we will apply the formula step by step.

### Step 1: Calculate \(a^2\)
\[ a^2 = (2x)^2 = 4x^2 \]

### Step 2: Calculate \(b^2\)
\[ b^2 = (3y)^2 = 9y^2 \]

### Step 3: Calculate \(c^2\)
\[ c^2 = (-4z)^2 = 16z^2 \]

### Step 4: Calculate \(2ab\)
\[ 2ab = 2 \cdot (2x) \cdot (3y) = 12xy \]

### Step 5: Calculate \(2bc\)
\[ 2bc = 2 \cdot (3y) \cdot (-4z) = -24yz \]

### Step 6: Calculate \(2ca\)
\[ 2ca = 2 \cdot (2x) \cdot (-4z) = -16zx \]

### Step 7: Combine all the results
Now we combine all the calculated parts:
\[ (2x + 3y - 4z)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca \]
\[ = 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16zx \]

Thus, the final expression is:
\[ (2x + 3y - 4z)^2 = 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16zx \]

### Final Answer:
The correct option is:
\[ 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16zx \]