Solution
= Solution
Take the <curl> component by component:
$$
\nabla\times\mathbf F=\left(\frac{\partial(3x+cy-2z)}{\partial y}-\frac{\partial(bx+y-z)}{\partial z},\ \frac{\partial(x+3y+az)}{\partial z}-\frac{\partial(3x+cy-2z)}{\partial x},\ \frac{\partial(bx+y-z)}{\partial x}-\frac{\partial(x+3y+az)}{\partial y}\right)=(c+1,a-3,b-3).
$$
The field is <irrotational> exactly when all three constant components vanish:
$$
\boxed{a=3,\qquad b=3,\qquad c=-1.}
$$