`ax^2+ by^2+ cz^2+2ayz+ 2bzx +2cxy` can be resolved into linear factors if a, b, c are such that

Explanation

Given expression is
`ax^(2)+by^(2)+cz^(2)+2ayz+2bzx+2cxy`……..i
`=z^(2)[a(x/z)^(2)+b(y/z)^(2)+c+2a(y/z)+2b(x/z)+2c(x/z)(y/z)]`
`=z^(2)[aX^(2)+bY^(2)+c+2aY+2bx+2cXY]`
where `x/z=X` and `y/z=Y`
Expression (i) will have two rational linear factors in x,y annd z if expression
`aX^(2)+bY^(2)+2cXY+2bx+2aY+c` will have two linear factors if
`abc+2abc-aa^(2)-bb^(2)-cc^(2)=0`
or `a^(3)+b^(3)+c^(3)=3abc`