1. **State the problem:** Simplify the expression $$(a-b)(a+b)^2(b-c)(b+c)^2(c-a)(c+a)^2$$. 2. **Recall the difference of squares formula:** $$x^2 - y^2 = (x-y)(x+y)$$ This formula will help us recognize patterns in the expression. 3. **Group the terms:** Group the factors as follows: $$(a-b)(a+b)^2, (b-c)(b+c)^2, (c-a)(c+a)^2$$ 4. **Rewrite each group:** Note that $$(a+b)^2 = (a+b)(a+b)$$, so $$(a-b)(a+b)^2 = (a-b)(a+b)(a+b) = (a^2 - b^2)(a+b)$$ Similarly, $$(b-c)(b+c)^2 = (b-c)(b+c)(b+c) = (b^2 - c^2)(b+c)$$ $$(c-a)(c+a)^2 = (c-a)(c+a)(c+a) = (c^2 - a^2)(c+a)$$ 5. **Substitute back:** The expression becomes $$(a^2 - b^2)(a+b)(b^2 - c^2)(b+c)(c^2 - a^2)(c+a)$$ 6. **Group the difference of squares terms:** $$ (a^2 - b^2)(b^2 - c^2)(c^2 - a^2) \times (a+b)(b+c)(c+a) $$ 7. **Recognize the product of differences:** Recall that $$ (a-b)(b-c)(c-a) $$ is related to the product of differences. 8. **Express $a^2 - b^2$ as $(a-b)(a+b)$:** $$a^2 - b^2 = (a-b)(a+b)$$ Similarly for the others. 9. **Rewrite the product:** $$ (a^2 - b^2)(b^2 - c^2)(c^2 - a^2) = (a-b)(a+b)(b-c)(b+c)(c-a)(c+a) $$ 10. **Substitute back:** The entire expression is now $$ (a-b)(a+b)(b-c)(b+c)(c-a)(c+a) \times (a+b)(b+c)(c+a) $$ 11. **Combine like terms:** Group the factors: $$ (a-b)(b-c)(c-a) \times (a+b)^2 (b+c)^2 (c+a)^2 $$ 12. **Final simplified form:** $$\boxed{(a-b)(b-c)(c-a)(a+b)^2(b+c)^2(c+a)^2}$$ This is the simplified expression, showing the product of differences and sums clearly.