Show that `Delta=|((y+z)^2,xy,zx),(xy,(x+z)^2,yz),(xz,yz,(x+y)^2)|=2xyz(x+y+z)^3`.

Explanation

`Given = |((y+z)^2,xy,zx),(xy,(x+z)^2,yz),(xz,yz,(x+y)^2)|=2xyz(x+y+z)^3`.


Applying `R_1 ->xR_1 , R_2 ->yR_2 , R_3 -> zR_3`


`|(x(y+z)^2,x^2y,zx^2),(xy^2,y(x+z)^2,y^2z),(xz^2,yz^2,z(x+y)^2)|`

`=(xyz)/(xyz) |((y+z)^2,x^2,x^2),(y^2,(x+z)^2,y^2),(z^2,z^2,(x+y)^2)|`
`c_2 -> c_2-c_1 . c_3-> c_3- c_1`
` |((y+z)^2,x^2-(y+z)^2,x^2- (y+z)^2),(y^2,(x+z)^2-y^2,0),(z^2,0,(x+y)^2-z^2)|`
Taking `(x+y+z)` common from `c_2` and `c_3`
` |((y+z)^2,x-y-z,x-y-z),(y^2,(x+z-y),0),(z^2,0,x+y-z)|`
`c_2`-> `c_2` + `(c_1)/y` and `c_3`-> `c_3` + `(c_1)/z`
`/_\=(x+y+z)^2 |(2yz,0,0),(y^2,x+z,y^2/z),(z^2,z^2/y,(x+y))|`
Expending along `R_1`
`= 2xyz(x+y+z)^3`