1. **Problem:** Factor the sum or difference of cubes for the expression $a^3 + 64$. 2. **Formula:** For sum of cubes, use $$x^3 + y^3 = (x + y)(x^2 - xy + y^2)$$ 3. **Identify terms:** Here, $a^3$ is $a^3$ and $64$ is $4^3$. 4. **Apply formula:** $$a^3 + 4^3 = (a + 4)(a^2 - 4a + 16)$$ 5. **Explanation:** We write the sum as cubes and then factor using the sum of cubes formula. The first factor is the sum of the cube roots, and the second factor is a trinomial with alternating signs. 6. **Final answer:** $$(a + 4)(a^2 - 4a + 16)$$