Express $x^2 y^2 + y^2 z^2 + z^2 x^2$ in the form of $x+y+z$, $xy+yz+zx$ and $xyz$?

Question

I'm not familiar with Symmetric polynomial. As title, is it possible to express $$x^2 y^2 + y^2 z^2 + z^2 x^2$$ using $x+y+z$, $xy+yz+zx$ and $xyz$?

Is there a trick for such problems?