Expand: `(2x + 3y - 4z)^(2)`

Answer

To expand the expression \((2x + 3y - 4z)^{2}\), we will use the formula for the square of a trinomial, which states:

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

### Step-by-Step Solution:

1. **Identify \(a\), \(b\), and \(c\)**:
- Let \(a = 2x\)
- Let \(b = 3y\)
- Let \(c = -4z\)

2. **Calculate \(a^{2}\)**:
\[ a^{2} = (2x)^{2} = 4x^{2} \]

3. **Calculate \(b^{2}\)**:
\[ b^{2} = (3y)^{2} = 9y^{2} \]

4. **Calculate \(c^{2}\)**:
\[ c^{2} = (-4z)^{2} = 16z^{2} \]

5. **Calculate \(2ab\)**:
\[ 2ab = 2 \cdot (2x) \cdot (3y) = 12xy \]

6. **Calculate \(2bc\)**:
\[ 2bc = 2 \cdot (3y) \cdot (-4z) = -24yz \]

7. **Calculate \(2ca\)**:
\[ 2ca = 2 \cdot (2x) \cdot (-4z) = -16xz \]

8. **Combine all the results**:
\[ (2x + 3y - 4z)^{2} = a^{2} + b^{2} + c^{2} + 2ab + 2bc + 2ca \]
\[ = 4x^{2} + 9y^{2} + 16z^{2} + 12xy - 24yz - 16xz \]

### Final Expanded Form:
\[ (2x + 3y - 4z)^{2} = 4x^{2} + 9y^{2} + 16z^{2} + 12xy - 24yz - 16xz \]