Using properties of determinants prove that:

Using properties of determinants prove that:

$\left|\begin{array}{ccc}a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+c y & 0\end{array}\right|=\left(b^{2}-a c\right)\left(a x^{2}+3 b x y+c y^{2}\right)$