Solve the following quantitative comparison problem by plugging in the number 0,1,2, -2, and `1//2` in that order - when possible.
`{:("Column A",x < 0,"Column B"),(x^2-x^5,,0):}`

Options

A. If column A is larger
B. If column B is larger
C. If the columns are equal
D. If there is not enough information to decide

Answer

To solve the quantitative comparison problem given, we will evaluate the expressions in Column A and Column B by substituting the values \(0, 1, 2, -2, \text{ and } -\frac{1}{2}\) in that order, while adhering to the condition \(x < 0\).

### Step-by-Step Solution:

1. **Identify the expressions:**
- Column A: \(x^2 - x^5\)
- Column B: \(0\)

2. **Determine valid substitutions:**
- Since we need \(x < 0\), we will only use negative values: \(-2, -1, -\frac{1}{2}\).

3. **Substituting \(x = -1\):**
- Calculate Column A:
\[ (-1)^2 - (-1)^5 = 1 - (-1) = 1 + 1 = 2 \]
- Compare with Column B:
\[ 2 > 0 \]
- Conclusion: Column A is greater than Column B.

4. **Substituting \(x = -2\):**
- Calculate Column A:
\[ (-2)^2 - (-2)^5 = 4 - (-32) = 4 + 32 = 36 \]
- Compare with Column B:
\[ 36 > 0 \]
- Conclusion: Column A is greater than Column B.

5. **Substituting \(x = -\frac{1}{2}\):**
- Calculate Column A:
\[ \left(-\frac{1}{2}\right)^2 - \left(-\frac{1}{2}\right)^5 = \frac{1}{4} - \left(-\frac{1}{32}\right) = \frac{1}{4} + \frac{1}{32} \]
- To add these fractions, find a common denominator (LCM of 4 and 32 is 32):
\[ \frac{1}{4} = \frac{8}{32} \quad \text{(since } 1 \times 8 = 8 \text{ and } 4 \times 8 = 32\text{)} \]
\[ \frac{8}{32} + \frac{1}{32} = \frac{9}{32} \]
- Compare with Column B:
\[ \frac{9}{32} > 0 \]
- Conclusion: Column A is greater than Column B.

6. **Final Conclusion:**
- In all cases where \(x < 0\) (i.e., \(x = -1, -2, -\frac{1}{2}\)), Column A is greater than Column B.

### Final Answer:
Column A is larger than Column B.