`{:("Column A",x = 1//y,"Column B"),((x^2 + x + 2)/x,,(2y^2 + y + 1)/(y)):}`

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 problem, we need to evaluate the expressions in Column A and Column B and determine their relationship.

### Step-by-Step Solution:

**Step 1: Analyze Column A**
We have the expression in Column A:
\[ \text{Column A} = \frac{x^2 + x + 2}{x} \]
Given that \( x = \frac{1}{y} \), we will substitute this value into the expression.

**Step 2: Substitute \( x \) in Column A**
Substituting \( x = \frac{1}{y} \) into Column A:
\[ \text{Column A} = \frac{\left(\frac{1}{y}\right)^2 + \left(\frac{1}{y}\right) + 2}{\frac{1}{y}} \]

**Step 3: Simplify the expression**
Calculating the numerator:
\[ \left(\frac{1}{y}\right)^2 = \frac{1}{y^2}, \quad \left(\frac{1}{y}\right) = \frac{1}{y}, \quad \text{and } 2 = \frac{2y^2}{y^2} \]
Thus, the numerator becomes:
\[ \frac{1}{y^2} + \frac{1}{y} + 2 = \frac{1 + y + 2y^2}{y^2} \]
Now substituting this back into Column A:
\[ \text{Column A} = \frac{\frac{1 + y + 2y^2}{y^2}}{\frac{1}{y}} = \frac{1 + y + 2y^2}{y^2} \cdot y = \frac{1 + y + 2y^2}{y} \]

**Step 4: Compare with Column B**
Now we look at Column B:
\[ \text{Column B} = \frac{2y^2 + y + 1}{y} \]

**Step 5: Conclusion**
We see that:
\[ \text{Column A} = \frac{1 + y + 2y^2}{y} \quad \text{and} \quad \text{Column B} = \frac{2y^2 + y + 1}{y} \]
Rearranging the terms in Column A gives us:
\[ \text{Column A} = \frac{2y^2 + y + 1}{y} \]
Thus, we find that:
\[ \text{Column A} = \text{Column B} \]

### Final Answer
Since both columns are equal, the correct option is:
**C: The columns are equal.**