Hidden symmetry

We could use $16925=25*677$ Then the polynomial factorises to $x^3+677x^2-25x-25*677=(x+677)(x+5)(x-5)$ So smallest Root is $-677$

$\displaystyle f(x,y)=(1+x+y^{2})(1+y+x^{2})=x^{3}+x^{2}(1+y^{2})+x(1+y)+y^{3}+y^{2}+y+1$

simply set $y=3i$ to get first expression and $y=-26$ to get the second.

The roots are $\sqrt{-1-y},-\sqrt{-1-y},-1-y^{2}$

It's written above 'hidden symmetry'. We can easily see (by using calculator) that 5 and -5 satisfy the eqn. Then by using sum of roots>>>>> 5+(-5)+c= -677 => c= -677

Seems a lot easier solving the equation directly, instead of trying to 1) figure out what "...look for some homogeneity" means, and 2) actually finding that "homogeneity", whatever that means. That seems like a very difficult way to solve such an equation.

How does that method help find roots of an equation?