(x+y)(y+z)(z+x)+xyz =

Expanding the terms, we get:

= (xy + xz + y^2 + yz)(z + x) + xyz
= xz^2 + x^2z + y^2z + xyz + xyz + x^2y + xyz + y^2y
= xz(z+x) + yz(z+x) + xy(z+x) + y^2z + x^2y
= xz^2 + x^2z + y^2z + y^2x + yz^2 + xyz
= z(x^2 + y^2 + xy) + y(xz + z^2) + x(y^2 + yz + z^2)
= z(x+y)(x+y) + y(z+x)(z+y) + x(y+z)(y+z)

Therefore, (x+y)(y+z)(z+x)+xyz = (x+y)(y+z)(z+x) + xyz = (x+y+z)(xy + xz + yz).