1. **State the problem:** Simplify the expression $$\frac{x^3 + x - 2}{x^3 + 1}$$ or analyze it. 2. **Recall formulas and rules:** - The denominator is a sum of cubes: $$x^3 + 1 = (x+1)(x^2 - x + 1)$$. - We can try to factor the numerator to see if any factors cancel. 3. **Factor the numerator:** Try to factor $$x^3 + x - 2$$ by checking possible roots using Rational Root Theorem. 4. **Check for roots:** Test $$x=1$$: $$1^3 + 1 - 2 = 1 + 1 - 2 = 0$$, so $$x=1$$ is a root. 5. **Divide numerator by $$x-1$$:** Using polynomial division or synthetic division: $$x^3 + x - 2 \div (x-1) = x^2 + x + 2$$ 6. **Rewrite numerator:** $$x^3 + x - 2 = (x-1)(x^2 + x + 2)$$ 7. **Rewrite the original expression:** $$\frac{(x-1)(x^2 + x + 2)}{(x+1)(x^2 - x + 1)}$$ 8. **Simplify:** No common factors to cancel, so the simplified form is: $$\frac{(x-1)(x^2 + x + 2)}{(x+1)(x^2 - x + 1)}$$ **Final answer:** $$\frac{(x-1)(x^2 + x + 2)}{(x+1)(x^2 - x + 1)}$$