1. **State the problem:** Simplify the expression $$ax^2 - a^3 - a^2 b + ab^2 + b^3 - b x^2$$. 2. **Group like terms:** Group terms with $x^2$ and those without: $$ax^2 - b x^2 - a^3 - a^2 b + ab^2 + b^3$$ 3. **Factor the $x^2$ terms:** $$x^2(a - b) - a^3 - a^2 b + ab^2 + b^3$$ 4. **Focus on the remaining terms:** $$- a^3 - a^2 b + ab^2 + b^3$$ 5. **Group and factor the remaining terms:** Group as $$(-a^3 - a^2 b) + (ab^2 + b^3)$$ Factor each group: $$-a^2(a + b) + b^2(a + b)$$ 6. **Factor out common factor $(a + b)$:** $$(a + b)(-a^2 + b^2)$$ 7. **Recognize difference of squares:** $$-a^2 + b^2 = (b - a)(b + a)$$ 8. **Rewrite the factorization:** $$(a + b)(b - a)(b + a) = (a + b)^2 (b - a)$$ 9. **Combine all parts:** $$x^2(a - b) + (a + b)^2 (b - a)$$ 10. **Note that $b - a = -(a - b)$, so:** $$x^2(a - b) - (a + b)^2 (a - b) = (a - b)(x^2 - (a + b)^2)$$ 11. **Recognize difference of squares again:** $$x^2 - (a + b)^2 = (x - (a + b))(x + (a + b))$$ 12. **Final factorization:** $$\boxed{(a - b)(x - (a + b))(x + (a + b))}$$