Using properties of determinants, prove the following `|[x,x+y,x+2y],[x+2y,x,x+y],[x+y,x+2y,x]|=9y^2(x+y)`

Explanation

`L.H.S. = |[x,x+y,x+2y],[x+2y,x,x+y],[x+y,x+2y,x]|`

Applying `R_1->R_1+R_2+R_3`

`= |[3(x+y),3(x+y),3(x+y)],[x+2y,x,x+y],[x+y,x+2y,x]|`

`=(3(x+y))|[1,1,1],[x+2y,x,x+y],[x+y,x+2y,x]|`

Now, applying `C_1->C_1-C_3 and C_2->C_2-C_3`

`=3(x+y)|[0,0,1],[y,-y,x+y],[y,2y,x]|`

`=3(x+y)[0-0+2^2+y^2]`

`=9y^2(x+y) = R.H.S.`