Find all critical points of $F(x,y) = (4y + x^2y^2 + 8x)/(xy) $

Question

Here is what I have so far :

$$F_x (x,y) = \frac{((2xy^2)(xy) - ((4y + x^2y^2 + 8x)(y)))}{(xy)^2} \\ = \frac{x^2y^3 -4y^2 + 8xy - 8x}{(xy)^2}$$

And : $$F_y (x,y) = \frac{((4 +2x^2y)(xy) - ((4y + x^2y^2 + 8x)(x)))}{(xy)^2} \\ = \frac{x^3y - 8x^2}{(xy)^2}$$

So I ended up using the quotient rule but this system seems difficult to solve is there a better way