1. **State the problem:** Simplify the expression $a^3 + a - 3a^2 - 3$. 2. **Group terms to factor:** Group the cubic and quadratic terms, and the linear and constant terms: $$a^3 - 3a^2 + a - 3$$ 3. **Factor by grouping:** $$ (a^3 - 3a^2) + (a - 3) $$ Factor out common terms in each group: $$ a^2(a - 3) + 1(a - 3) $$ 4. **Factor out the common binomial:** $$ (a - 3)(a^2 + 1) $$ 5. **Final answer:** The expression factors to $$ (a - 3)(a^2 + 1) $$ This is the simplified factorization of the original expression.