Solution

Applying \(R_1 \to R_1 - xR_2\), we get \(\Delta = |\begin{smallmatrix} a(1-x^2) & c(1-x^2) & p(1-x^2) \\ ax+b & cx+d & px+q \\ u & v & w \end{smallmatrix}|\) \(= (1-x^2)|\begin{smallmatrix} a & c & p \\ ax+b & cx+d & px+q \\ u & v & w \end{smallmatrix}|\) Applying \(R_2 \to R_2 - xR_1\), we get \(\Delta = (1-x^2)|\begin{smallmatrix} a & c & p \\ b & d & q \\ u & v & w \end{smallmatrix}|\)