Let `Sn = n^2 + 20n + 12` where n is a positive integer. What is the sum of all possible values of n for which `S_(n)` is a perfect square ?

Answer

To solve the problem, we need to find the values of \( n \) such that \( S_n = n^2 + 20n + 12 \) is a perfect square. Let's denote the perfect square by \( y^2 \).

### Step-by-step Solution:

1. **Set up the equation**:
\[ S_n = n^2 + 20n + 12 = y^2 \]

2. **Rearranging the equation**:
\[ n^2 + 20n + 12 - y^2 = 0 \]
This is a quadratic equation in \( n \).

3. **Complete the square**:
We can rewrite the quadratic expression by completing the square:
\[ n^2 + 20n + 100 - 100 + 12 = y^2 \]
\[ (n + 10)^2 - y^2 = 88 \]
Thus, we have:
\[ (n + 10)^2 - y^2 = 88 \]

4. **Factor the difference of squares**:
\[ (n + 10 - y)(n + 10 + y) = 88 \]
Let \( a = n + 10 - y \) and \( b = n + 10 + y \). Then \( ab = 88 \).

5. **Find pairs of factors of 88**:
The pairs of factors of 88 are:
- \( (1, 88) \)
- \( (2, 44) \)
- \( (4, 22) \)
- \( (8, 11) \)

6. **Set up equations for each factor pair**:
For each pair \( (a, b) \):
\[ a + b = 2(n + 10) \quad \text{and} \quad b - a = 2y \]

- **Case 1**: \( a = 1, b = 88 \)
\[ 1 + 88 = 2(n + 10) \Rightarrow 89 = 2(n + 10) \Rightarrow n + 10 = 44.5 \Rightarrow n = 34.5 \quad \text{(not an integer)} \]

- **Case 2**: \( a = 2, b = 44 \)
\[ 2 + 44 = 2(n + 10) \Rightarrow 46 = 2(n + 10) \Rightarrow n + 10 = 23 \Rightarrow n = 13 \quad \text{(valid integer)} \]

- **Case 3**: \( a = 4, b = 22 \)
\[ 4 + 22 = 2(n + 10) \Rightarrow 26 = 2(n + 10) \Rightarrow n + 10 = 13 \Rightarrow n = 3 \quad \text{(valid integer)} \]

- **Case 4**: \( a = 8, b = 11 \)
\[ 8 + 11 = 2(n + 10) \Rightarrow 19 = 2(n + 10) \Rightarrow n + 10 = 9.5 \Rightarrow n = -0.5 \quad \text{(not an integer)} \]

7. **Collect valid integer solutions**:
The valid integer solutions we found are \( n = 13 \) and \( n = 3 \).

8. **Calculate the sum of all possible values of \( n \)**:
\[ \text{Sum} = 13 + 3 = 16 \]

### Final Answer:
The sum of all possible values of \( n \) for which \( S_n \) is a perfect square is \( \boxed{16} \).