1. **State the problem:** We need to find the quotient of the functions $f(x) = x^3 + x^2 + 2$ and $g(x) = x - 1$, i.e., compute $\frac{f(x)}{g(x)}$. 2. **Formula and method:** To divide polynomials, we perform polynomial long division or synthetic division. Here, we divide $f(x)$ by $g(x)$. 3. **Set up the division:** Divide $x^3 + x^2 + 0x + 2$ by $x - 1$. 4. **Perform the division:** - Divide the leading term $x^3$ by $x$ to get $x^2$. - Multiply $x^2$ by $x - 1$ to get $x^3 - x^2$. - Subtract: $(x^3 + x^2) - (x^3 - x^2) = 2x^2$. - Bring down the next term $0x$. - Divide $2x^2$ by $x$ to get $2x$. - Multiply $2x$ by $x - 1$ to get $2x^2 - 2x$. - Subtract: $(2x^2 + 0x) - (2x^2 - 2x) = 2x$. - Bring down the last term $+2$. - Divide $2x$ by $x$ to get $2$. - Multiply $2$ by $x - 1$ to get $2x - 2$. - Subtract: $(2x + 2) - (2x - 2) = 4$. 5. **Result:** The quotient is $x^2 + 2x + 2$ with a remainder of $4$. 6. **Express the division:** $$\frac{f(x)}{g(x)} = x^2 + 2x + 2 + \frac{4}{x - 1}$$ **Final answer:** $$\boxed{\frac{x^3 + x^2 + 2}{x - 1} = x^2 + 2x + 2 + \frac{4}{x - 1}}$$