\textbf{F}=(ax^2+bxy+cy^2-2x)\textbf{i}+(x^2+xy-y^2+bz)\textbf{j}+(2y+2z)\textbf{k}

I must show that the above equation is irrotational and solinoidal (ie; satisfies the Laplace Equation) for chosen values of a, b and c.

I have managed to show its solinoidal and irrotational seperatly for different values of a, b and c, but I don't know how to show both properties at the same time. Is there a condition I must satisfy?