1. **State the problem:** Simplify or factor the expression $a^3 + c^3$ given the sum $a + c$. 2. **Formula used:** The sum of cubes factorization formula is: $$a^3 + c^3 = (a + c)(a^2 - ac + c^2)$$ This formula states that the sum of two cubes can be factored into the product of the sum of the bases and a quadratic expression. 3. **Apply the formula:** Since we have $a + c$ already, we can write: $$a^3 + c^3 = (a + c)(a^2 - ac + c^2)$$ 4. **Explanation:** This means that the cubic sum breaks down into a linear factor $(a + c)$ and a quadratic factor $(a^2 - ac + c^2)$. 5. **Intermediate step:** If you want to verify, you can expand the right side: $$\begin{aligned} (a + c)(a^2 - ac + c^2) &= a \cdot a^2 - a \cdot ac + a \cdot c^2 + c \cdot a^2 - c \cdot ac + c \cdot c^2 \\ &= a^3 - a^2 c + a c^2 + a^2 c - a c^2 + c^3 \\ &= a^3 + c^3 \quad \text{(since } -a^2 c + a^2 c = 0 \text{ and } a c^2 - a c^2 = 0) \end{aligned}$$ 6. **Final answer:** $$a^3 + c^3 = (a + c)(a^2 - ac + c^2)$$