$$a^{2}(b^{2}-c^{2})+b^{2}(c^{2}-a^{2})+c^{2}(a^{2}-b^{2})$$
$0$
$$a^{2}b^{2}-a^{2}c^{2}+b^{2}\left(c^{2}-a^{2}\right)+c^{2}\left(a^{2}-b^{2}\right)$$
$$a^{2}b^{2}-a^{2}c^{2}+b^{2}c^{2}-b^{2}a^{2}+c^{2}\left(a^{2}-b^{2}\right)$$
$$-a^{2}c^{2}+b^{2}c^{2}+c^{2}\left(a^{2}-b^{2}\right)$$
$$-a^{2}c^{2}+b^{2}c^{2}+c^{2}a^{2}-c^{2}b^{2}$$
$$b^{2}c^{2}-c^{2}b^{2}$$
$$0$$
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