The expression $a^3 + 1$ can be factored using the sum of cubes formula:

$a^3 + 1 = (a + 1)(a^2 - a + 1)$

This is the factorized form of the expression.

Would you like more details or have any questions?

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Here are 8 related questions you might find interesting:

1. How do you factor the difference of cubes, $a^3 - b^3$?
2. What are the roots of the equation $a^3 + 1 = 0$?
3. Can you solve $a^3 + b^3 = 0$ using the sum of cubes formula?
4. How can the expression $a^3 + b^3 + c^3$ be factored?
5. What is the geometric interpretation of the sum of cubes?
6. How does the factorization of $a^3 + 1$ differ when $a$ is a complex number?
7. What happens when you set $a = 1$ in the expression $a^3 + 1$?
8. Can you derive the sum of cubes formula from the expansion of $(a + b)(a^2 - ab + b^2)$?

Tip: When factoring cubic expressions, always look for patterns like the sum or difference of cubes, which have standard formulas.