We have to prove the following result without expanding
$\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} \right|=(ab+bc+ca)\left|\begin{array}{lll} a^2 & a &1 \\ b^2 & b &1\\ c^2 & c &1 \end{array} \right|$
Progress :
$\left|\begin{array}{lll} a^3 & a^2 &1 \\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} \right|=\frac{1}{abc}\left|\begin{array}{lll} a^3 & a^2 &abc \\ b^3 & b^2 &abc\\ c^3 & c^2 &abc \end{array} \right|=\left|\begin{array}{lll} a^2 & a &ab \\ b^2 & b & ca\\ c^2 & c &ab \end{array} \right|=\left|\begin{array}{ccc} a^2+b^2+c^2 & a+b+c &ab+bc+ca\\ b^3 & b^2 &1\\ c^3 & c^2 &1 \end{array} \right|$
What will be the next
