Answer
Not Conservative
Work Step by Step
As we are given that $F=(z+y) i+zj+(y+x) k$
Definition of curl F is: $\text{curl} F =(\dfrac{\partial R}{\partial y}-\dfrac{\partial Q}{\partial z})i +(\dfrac{\partial P}{\partial z}-\dfrac{\partial R}{\partial x}) k+(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y})k $
A vector field is conservative iff the $\text{curl} F =0$
Now, curl F$=(1-1) i+(1-1)j +(0-1) k=-k \ne 0$
Hence, the vector field is Not Conservative