C

To find the cross product \( \mathbf{A} \times \mathbf{B} \), we can use the determinant method with unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).
\A = 2\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}
\B = 4\mathbf{i} + 3\mathbf{j} + 2\mathbf{k}
The determinant setup is:
\[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 4 & 3 & 2 \end{vmatrix} \]
Expanding this determinant, we have:
\[ \mathbf{A} \times \mathbf{B} = \mathbf{i}(3 \cdot 2 - 4 \cdot 3) - \mathbf{j}(2 \cdot 2 - 4 \cdot 4) + \mathbf{k}(2 \cdot 3 - 3 \cdot 4)
\]
Calculating each component:
\[ = \mathbf{i}(6 - 12) - \mathbf{j}(4 - 16) + \mathbf{k}(6 - 12)
\]
\[ = -6\mathbf{i} + 12\mathbf{j} - 6\mathbf{k}
\]
Thus, \( \mathbf{A} \times \mathbf{B} = -6\mathbf{i} + 12\mathbf{j} - 6\mathbf{k} \)
Therefore, the final result corresponds to option C.