If $x+y+z=0$, then $(x+y)(y+z)(z+x)+xyz$ equals:

From $x+y+z=0$ each bracket is the negative of the missing letter: $x+y=-z,y+z=-x$ and $z+x=-y$. Their product is $(-z)(-x)(-y)=-xyz$, so the whole expression is $-xyz+xyz=0.(xyz$ and 2$xyz$ come from missing the minus sign that three negative factors produce; $-xyz$ forgets to add the final term.) Test $x,y,z=2,3,-5:(5)(-2)(-3)+2(3)(-5)=30-30=0$.