1. **State the problem:** Simplify the expression $$(x - 1)(x + 2) \sqrt{x^4 + 3x^3 - x^2 + x + 2}$$ 2. **Recall the formula and rules:** - Multiply binomials using the distributive property (FOIL method). - Simplify the polynomial inside the square root if possible. - The square root of a product is the product of the square roots if all terms are non-negative. 3. **Multiply the binomials:** $$(x - 1)(x + 2) = x \cdot x + x \cdot 2 - 1 \cdot x - 1 \cdot 2 = x^2 + 2x - x - 2 = x^2 + x - 2$$ 4. **Analyze the polynomial inside the square root:** $$x^4 + 3x^3 - x^2 + x + 2$$ Try to factor it or check if it can be simplified. 5. **Attempt to factor the quartic polynomial:** Try factoring by grouping or synthetic division. 6. **Check for rational roots using Rational Root Theorem:** Possible roots: $\pm1, \pm2$ 7. **Test $x = -1$:** $$(-1)^4 + 3(-1)^3 - (-1)^2 + (-1) + 2 = 1 - 3 - 1 - 1 + 2 = -2 \neq 0$$ 8. **Test $x = 1$:** $$1 + 3 - 1 + 1 + 2 = 6 \neq 0$$ 9. **Test $x = -2$:** $$16 - 24 - 4 - 2 + 2 = -12 \neq 0$$ 10. **Test $x = 2$:** $$16 + 24 - 4 + 2 + 2 = 40 \neq 0$$ No rational roots found, so the polynomial is not easily factorable. 11. **Conclusion:** The expression simplifies to: $$\boxed{(x^2 + x - 2) \sqrt{x^4 + 3x^3 - x^2 + x + 2}}$$ This is the simplest exact form without further factorization of the quartic polynomial inside the square root.