If $$x + y$$  and $$y + 3x$$  are two factors of the expression $$\lambda {x^3} - \mu {x^2}y + x{y^2} + {y^3}$$     then the third factor is

Solution :
As it is a third-degree homogeneous expression in $$x, y,$$  we have $${y^3} + {y^2}x - \mu y{x^2} + \lambda {x^3}$$
$$\eqalign{ & = \left( {y + x} \right)\left( {y + 3x} \right)\left( {y + mx} \right) \cr & = {y^3} + \left( {m + 3 + 1} \right){y^2}x + \left( {3 + m + 3m} \right)y{x^2} + 3m{x^3} \cr & \Rightarrow \,\,1 = m + 4, - \mu = 3 + 4m,\lambda = 3m. \cr} $$