1. **State the problem:** We are given the volume of a rectangular box as $x^3 + 3x^2 + 2x - 5$ cubic centimeters and the height as $x + 1$ centimeters. We need to find the area of the base, which means dividing the volume by the height. 2. **Set up the polynomial division:** We will divide $$x^3 + 3x^2 + 2x - 5$$ by $$x + 1$$ using long division. 3. **Divide the leading terms:** Divide $x^3$ by $x$ to get $x^2$. This is the first term of the quotient. 4. **Multiply and subtract:** Multiply $x^2$ by $x + 1$ to get $x^3 + x^2$. Subtract this from the original polynomial: $$ (x^3 + 3x^2 + 2x - 5) - (x^3 + x^2) = (3x^2 - x^2) + 2x - 5 = 2x^2 + 2x - 5 $$ 5. **Next division step:** Divide the leading term $2x^2$ by $x$ to get $2x$. Add $2x$ to the quotient. 6. **Multiply and subtract:** Multiply $2x$ by $x + 1$ to get $2x^2 + 2x$. Subtract this from the remainder: $$ (2x^2 + 2x - 5) - (2x^2 + 2x) = -5 $$ 7. **Final division step:** Divide the leading term of remainder $-5$ by $x$. Since $-5$ is a constant and degree of divisor is 1, the division stops here with remainder $-5$. 8. **Conclusion:** The quotient is $x^2 + 2x$ with a remainder of $-5$, so the exact polynomial division is: $$ \frac{x^3 + 3x^2 + 2x - 5}{x + 1} = x^2 + 2x + \frac{-5}{x + 1} $$ Since the area of the base must be a polynomial (height is a factor), the remainder indicates that the height $x + 1$ is not an exact factor of the volume. 9. **Interpretation:** If the problem assumes exact division, the calculation suggests a possible typo. If we consider only the quotient ignoring the remainder as the approximate area of base, then: **Area of base** $= x^2 + 2x$ This is the polynomial representing the area of the base in square centimeters.