If `A={p:p=((n+2)(2n^(5)+3n^(4)+4n^(3)+5n^(2)+6))/(n^(2)+2n), n p in Z^(+)}` then the number of elements in the set A, is

Options

A. 2
B. 3
C. 4
D. 6

Answer

To solve the problem, we need to analyze the expression given for \( P \) and determine how many distinct positive integer values it can take based on the values of \( n \).

### Step-by-Step Solution:

1. **Understanding the Expression for \( P \)**:
The expression for \( P \) is given as:
\[ P = \frac{(n + 2)(2n^5 + 3n^4 + 4n^3 + 5n^2 + 6)}{n^2 + 2n} \]
We need to simplify this expression.

**Hint**: Look for common factors in the numerator and denominator.

2. **Simplifying the Expression**:
We can factor the denominator:
\[ n^2 + 2n = n(n + 2) \]
Now, we can cancel \( (n + 2) \) from the numerator and denominator (provided \( n \neq -2 \), which is not an issue since \( n \) is a positive integer):
\[ P = \frac{2n^5 + 3n^4 + 4n^3 + 5n^2 + 6}{n} \]
This simplifies to:
\[ P = 2n^4 + 3n^3 + 4n^2 + 5n + \frac{6}{n} \]

**Hint**: Ensure that \( \frac{6}{n} \) is an integer for \( P \) to be a positive integer.

3. **Finding Values of \( n \)**:
For \( P \) to be a positive integer, \( \frac{6}{n} \) must also be a positive integer. This means \( n \) must be a divisor of 6. The positive divisors of 6 are:
\[ n = 1, 2, 3, 6 \]

**Hint**: List all positive divisors of 6.

4. **Calculating \( P \) for Each Valid \( n \)**:
Now we will calculate \( P \) for each divisor:
- For \( n = 1 \):
\[ P = 2(1^4) + 3(1^3) + 4(1^2) + 5(1) + 6 = 2 + 3 + 4 + 5 + 6 = 20 \]
- For \( n = 2 \):
\[ P = 2(2^4) + 3(2^3) + 4(2^2) + 5(2) + 3 = 32 + 24 + 16 + 10 + 3 = 85 \]
- For \( n = 3 \):
\[ P = 2(3^4) + 3(3^3) + 4(3^2) + 5(3) + 2 = 162 + 81 + 36 + 15 + 2 = 296 \]
- For \( n = 6 \):
\[ P = 2(6^4) + 3(6^3) + 4(6^2) + 5(6) + 1 = 2592 + 648 + 144 + 30 + 1 = 3415 \]

**Hint**: Ensure to substitute each \( n \) back into the simplified expression for \( P \).

5. **Identifying Distinct Values of \( P \)**:
The values of \( P \) we calculated are:
- For \( n = 1 \): \( P = 20 \)
- For \( n = 2 \): \( P = 85 \)
- For \( n = 3 \): \( P = 296 \)
- For \( n = 6 \): \( P = 3415 \)

All these values are distinct.

6. **Counting the Elements in Set \( A \)**:
Since we have found 4 distinct values of \( P \), the number of elements in the set \( A \) is:
\[ \text{Number of elements in } A = 4 \]

### Final Answer:
The number of elements in the set \( A \) is **4**.