I have to prove that by rotation of axes the expression $F(x,y,z) = ax^2 + by^2 + cz^2 +2fyz + 2gzx + 2hxy $ transforms to $ λ_1x^2 + λ_2y^2 + λ_3z^2 =0 $ where $λ_1, λ_2,λ_3 $ are roots of cubic
$ \begin{vmatrix} a-λ&h&g\\ h&b-λ&f\\ g&f&c-λ\\ \end{vmatrix}=0 $
Solution in book says that $ax^2 + by^2 + cz^2 +2fyz + 2gzx + 2hxy - λ(x^2 +y^2+z^2)$ should reduce to $ λ_1x^2 + λ_2y^2 + λ_3z^2-λ(x^2 +y^2+z^2)$ . Hence, book says, both these expressions will be product of linear factors for same value of λ. I don't understand why this is necessary.
Further, it is said that if $(a-λ)x^2 + (b-λ)y^2 + (c-λ)z^2 +2fyz + 2gzx + 2hxy $ is a product of two linear factors then $ \begin{vmatrix} a-λ&h&g\\ h&b-λ&f\\ g&f&c-λ\\ \end{vmatrix}=0 $
I also don't get how this condition is arrived at. Any help in clearing these two doubts will be helpful.