Using the properties of determinants, prove that following `|(a-b,-c^2,a^2),(a^2,-c,-a^2),(b^2,c^2,-a-b)|=(a+b+c)^3`

Explanation

L.H.S. $$\Delta=\left|\begin{array}{ccc}a^{2}+1 & a b & a c \\ b a & b^{2}+1 & b c \\ c a & c b & c^{2}+1\end{array}\right|$$
Operating $$\mathrm{R}_{1} \rightarrow \frac{1}{a} \mathrm{R}_{1}, \mathrm{R}_{2} \rightarrow \frac{1}{b} \mathrm{R}_{2}$$ and $$\mathrm{R}_{3} \rightarrow \frac{1}{c} \mathrm{R}_{3}$$, we have
$$
\Delta=a b c\left|\begin{array}{ccc}
a+\frac{1}{a} & b & c \\
a & b+\frac{1}{b} & c \\
a & b & c+\frac{1}{c}
\end{array}\right|
...