1. **State the problem:** Factor the cubic polynomial $$a^3 - 3a + 2$$ completely. 2. **Recall the formula and rules:** To factor a cubic polynomial, we can try to find rational roots using the Rational Root Theorem, then use polynomial division or factor by grouping. 3. **Find possible rational roots:** Factors of the constant term 2 are $$\pm1, \pm2$$. 4. **Test roots:** - For $$a=1$$: $$1^3 - 3(1) + 2 = 1 - 3 + 2 = 0$$, so $$a=1$$ is a root. 5. **Divide the polynomial by $a-1$:** Use synthetic division: $$\begin{array}{r|rrrr} 1 & 1 & 0 & -3 & 2 \\ & & 1 & 1 & -2 \\ \hline & 1 & 1 & -2 & 0 \end{array}$$ The quotient is $$a^2 + a - 2$$. 6. **Factor the quadratic:** $$a^2 + a - 2 = (a + 2)(a - 1)$$. 7. **Write the full factorization:** $$a^3 - 3a + 2 = (a - 1)(a + 2)(a - 1) = (a - 1)^2 (a + 2)$$. **Final answer:** $$\boxed{(a - 1)^2 (a + 2)}$$