<s>[INST] Calculate: $\Delta=\left|\begin{array}{ccc} b c & c a & a b \\ a(b+c) & b(c+a) & c(a+b) \\ a^{2} & b^{2} & c^{2} \end{array}\right|$ Calculate:
$$\Delta=\left|\begin{array}{ccc}
b c & c a & a b \\
a(b+c) & b(c+a) & c(a+b) \\
a^{2} & b^{2} & c^{2}
\end{array}\right|$$
Does anyone know any easy way to calculate this determinant?
I tried the classic way but I was wondering if anyone knows an easier method.
 [/INST] \begin{gathered}
\left|\begin{array}{ccc}
b c & a c & a b \\
a b+a c & a b+b c & a c+b c \\
a^{2} & b^{2} & c^{2}
\end{array}\right|=b c \cdot(a b+b c) \cdot c^{2}+a c \cdot(a c+b c) \cdot a^{2}+a b \cdot(a b+a c) \cdot b^{2}-a^{2} \cdot(a b+b c) \cdot(a b)-b^{2} \cdot(a c+b c) \cdot(b c)-c^{2} \cdot(a b+a c) \cdot(a c) \\
\equiv \\
=a^{2} b^{4}-a^{4} b^{2}-a^{2} c^{4}+b^{2} c^{4}+a b^{2} c^{3}-a^{2} b c^{3}+a^{4} c^{2}-b^{4} c^{2}-a b^{3} c^{2}+a^{3} b c^{2}+a^{2} b^{3} c-a^{3} b^{2} c
\end{gathered}
</s>