1. The problem is to divide the polynomial $x^3 - 2$ by $x^2 + 1$ using long division. 2. Set up the long division: divide the first term of the dividend $x^3$ by the first term of the divisor $x^2$ to get $x$. 3. Multiply the entire divisor $x^2 + 1$ by $x$ to get $x^3 + x$. 4. Subtract this from the dividend: $(x^3 - 2) - (x^3 + x) = -x - 2$. 5. Now, divide the first term of the new dividend $-x$ by the first term of the divisor $x^2$. Since the degree of $-x$ is less than the degree of $x^2$, division stops here. 6. Therefore, the quotient is $x$ and the remainder is $-x - 2$. 7. The final answer is written as: $$x^3 - 2 \div (x^2 + 1) = x + \frac{-x - 2}{x^2 + 1}$$