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Prove that
$\Delta=\left|\begin{array}{ccc}
a+b x & c+d x & p+q x \\
a x+b & c x+d & p x+q \\
u & v & w
\end{array}\right|=\left(1-x^{2}\right)\left|\begin{array}{lll}
a & c & p \\
b & d & q \\
u & v & m
\end{array}\right|$
Solution
Applying $\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-x \mathrm{R}_{2}$ to $\Delta,$ we get
${\Delta = \left| {\begin{array}{*{20}{c}}
{a\left( {1 – {x^2}} \right)}&{c\left( {1 – {x^2}} \right)}&{p\left( {1 – {x^2}} \right)} \\
{ax + b}&{cx + d}&{px + q} \\
u&v&w
\end{array}} \right|}$
${ = \left( {1 – {x^2}} \right)\left| {\begin{array}{*{20}{c}}
a&c&p \\
{ax + b}&{cx + d}&{px + q} \\
u&v&w
\end{array}} \right|}$
Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}-x \mathrm{R}_{1},$ we get
$\Delta=\left(1-x^{2}\right)\left|\begin{array}{lll}
a & c & p \\
b & d & q \\
u & v & w
\end{array}\right|$
