1. **State the problem:** Find the product of the polynomials $$(x^3 + 2x^2 - x - 2)(x^2 + x - 1).$$ 2. **Recall the distributive property:** To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and then combine like terms. 3. **Multiply each term:** - Multiply $x^3$ by each term in $x^2 + x - 1$: $$x^3 \cdot x^2 = x^5,$$ $$x^3 \cdot x = x^4,$$ $$x^3 \cdot (-1) = -x^3.$$ - Multiply $2x^2$ by each term: $$2x^2 \cdot x^2 = 2x^4,$$ $$2x^2 \cdot x = 2x^3,$$ $$2x^2 \cdot (-1) = -2x^2.$$ - Multiply $-x$ by each term: $$-x \cdot x^2 = -x^3,$$ $$-x \cdot x = -x^2,$$ $$-x \cdot (-1) = +x.$$ - Multiply $-2$ by each term: $$-2 \cdot x^2 = -2x^2,$$ $$-2 \cdot x = -2x,$$ $$-2 \cdot (-1) = +2.$$ 4. **Write all terms together:** $$x^5 + x^4 - x^3 + 2x^4 + 2x^3 - 2x^2 - x^3 - x^2 + x - 2x^2 - 2x + 2.$$ 5. **Combine like terms:** - Combine $x^4$ terms: $$x^4 + 2x^4 = 3x^4.$$ - Combine $x^3$ terms: $$-x^3 + 2x^3 - x^3 = 0.$$ - Combine $x^2$ terms: $$-2x^2 - x^2 - 2x^2 = -5x^2.$$ - Combine $x$ terms: $$x - 2x = -x.$$ - Constant term: $$2.$$ 6. **Final simplified product:** $$\boxed{x^5 + 3x^4 - 5x^2 - x + 2}.$$