1. **Problem statement:** Solve the equation $$(x - 1)^3 + 3(x + 1)^2 + x(x - 3)(x + 3) = 2x^3 + 2.$$ 2. **Recall formulas and rules:** - Expand powers and products carefully. - Use the identity $x^2 - 9 = (x - 3)(x + 3)$. - Combine like terms and simplify both sides. 3. **Expand each term:** - $(x - 1)^3 = x^3 - 3x^2 + 3x - 1$ - $(x + 1)^2 = x^2 + 2x + 1$ so $3(x + 1)^2 = 3x^2 + 6x + 3$ - $x(x - 3)(x + 3) = x(x^2 - 9) = x^3 - 9x$ 4. **Sum left side:** $$x^3 - 3x^2 + 3x - 1 + 3x^2 + 6x + 3 + x^3 - 9x = (x^3 + x^3) + (-3x^2 + 3x^2) + (3x + 6x - 9x) + (-1 + 3)$$ $$= 2x^3 + 0 + 0 + 2 = 2x^3 + 2$$ 5. **Right side is already:** $$2x^3 + 2$$ 6. **Set equation:** $$2x^3 + 2 = 2x^3 + 2$$ 7. **Conclusion:** Both sides are equal for all real $x$. Therefore, the equation holds for every real number. **Final answer:** All real numbers $x$ satisfy the equation.