Determine whether the following vector field is conservative on $\mathbf{R}^{3}$. If so, determine the potential function.
\[
F=\langle 5 y+4 z, 5 x+2 z, 4 x+2 y\rangle
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. $\mathbf{F}$ is conservative on $\mathbf{R}^{3}$. Assuming the arbitrary constant is 0 , the potential function is $\varphi(x, y, z)=$
B. $\mathbf{F}$ is not conservative on $\mathbf{R}^{3}$.

Step 1 ：Given the vector field \(F=\langle 5 y+4 z, 5 x+2 z, 4 x+2 y\rangle\), we need to determine whether it is conservative on \(\mathbf{R}^{3}\).

Step 2 ：A vector field is conservative if its curl is zero. The curl of a vector field \(F = \langle P, Q, R \rangle\) is given by the determinant of the following matrix: \[ \begin{vmatrix} i & j & k \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \]

Step 3 ：Let's calculate the curl of the given vector field. Here, \(P = 5y + 4z\), \(Q = 5x + 2z\), and \(R = 4x + 2y\).

Step 4 ：The curl of the vector field is \(-4, 0, -3\).

Step 5 ：Since the curl of the vector field is not zero, the vector field is not conservative.

Step 6 ：\(\boxed{\text{The vector field } F \text{ is not conservative on } \mathbf{R}^{3}}\)