1. **State the problem:** Simplify the expression $$a (a^2 - a^2) + a (a^2 + a^2) + a (-a^2 - a^2)$$. 2. **Recall the distributive property:** For any terms $x$, $y$, and $z$, $x(y + z) = xy + xz$. 3. **Apply the distributive property to each term:** - First term: $$a (a^2 - a^2) = a \cdot a^2 - a \cdot a^2 = a^3 - a^3$$ - Second term: $$a (a^2 + a^2) = a \cdot a^2 + a \cdot a^2 = a^3 + a^3$$ - Third term: $$a (-a^2 - a^2) = a \cdot (-a^2) + a \cdot (-a^2) = -a^3 - a^3$$ 4. **Combine all terms:** $$a^3 - a^3 + a^3 + a^3 - a^3 - a^3$$ 5. **Simplify by combining like terms:** $$= (a^3 - a^3) + (a^3 + a^3) + (-a^3 - a^3) = 0 + 2a^3 - 2a^3 = 0$$ **Final answer:** $$0$$