$(b - c)(b + c)^2 + (c - a)(c + a)^2 + (a - b)(a + b)^2 = -(a - b)(b - c)(c - a)$
$((b - c)(b + c))(b + c) + ((c - a)(c + a))(c + a) + ((a - b)(a + b))(a + b) = -(ab - b^2 - ac + bc)(c - a)$
$(b^2 - c^2)(b + c) + (c^2 - a^2)(c + a) + (a^2 - b^2)(a + b) = -(abc - b^2c - ac^2 + bc^2 - a^2b + ab^2 + a^2c - abc)$
$b^3 + b^2c - bc^2 - c^3 + c^3 - a^2c + ac^2 - a^3 + a^3 - ab^2 + a^2b - b^3 = -(-b^2c - ac^2 + bc^2 - a^2b + ab^2 + a^2c)$
$b^2c - bc^2 - a^2c + ac^2 - ab^2 + a^2b = b^2c - bc^2 - a^2c + ac^2 - ab^2 + a^2b$