1. The problem involves expressions with sums of pairs of variables: $a+b$, $b+c$, and $c+a$. 2. We are given the expression $b+c, c+a, a+b = 2 b, c, a = 2(3abc - a^3 - b^3 - c^3)$ and need to understand or simplify it. 3. Let's clarify the expression: it seems to relate the products of sums of pairs to a formula involving $abc$ and cubes of $a,b,c$. 4. Recall the identity for the product of sums of pairs: $$ (a+b)(b+c)(c+a) = (a+b+c)(ab+bc+ca) - abc $$ 5. Also, the sum of cubes identity: $$ a^3 + b^3 + c^3 - 3abc = (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) $$ 6. The expression $2(3abc - a^3 - b^3 - c^3)$ can be rewritten as: $$ 2(3abc - (a^3 + b^3 + c^3)) = 2(3abc - a^3 - b^3 - c^3) $$ 7. Using the sum of cubes identity, we have: $$ 3abc - (a^3 + b^3 + c^3) = - (a^3 + b^3 + c^3 - 3abc) = - (a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) $$ 8. Therefore: $$ 2(3abc - a^3 - b^3 - c^3) = -2(a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) $$ 9. This shows the right side is related to symmetric sums of $a,b,c$. 10. The problem likely asks to verify or simplify the given relation. Final answer: $$ (a+b)(b+c)(c+a) = (a+b+c)(ab+bc+ca) - abc $$ and $$ 2(3abc - a^3 - b^3 - c^3) = -2(a+b+c)(a^2 + b^2 + c^2 - ab - bc - ca) $$ which are key identities involving these expressions.