A question on polynomials

Let $f(x,y) = ({x^2} + {y^2})p{(x,y)^2} – q{(x,y)^2}$ and $p$ and $q$ are two polynomials.

Is it true that;

$frac{{partial f}}{{partial x}}$ and $frac{{partial f}}{{partial y}}$ don’t have a common factor?