Find the sum of all those integers n for which `n^2+20n+15` is the square of an integer.

Answer

To find the sum of all integers \( n \) for which \( n^2 + 20n + 15 \) is a perfect square, we can follow these steps:

### Step 1: Set up the equation
We want to find integers \( n \) such that:
\[ n^2 + 20n + 15 = m^2 \]
where \( m \) is also an integer.

### Step 2: Rearrange the equation
Rearranging gives us:
\[ n^2 + 20n + 15 - m^2 = 0 \]
This is a quadratic equation in \( n \).

### Step 3: Complete the square
To make it easier to solve, we complete the square for the \( n \) terms:
\[ n^2 + 20n = (n + 10)^2 - 100 \]
Thus, we can rewrite the equation as:
\[ (n + 10)^2 - 100 + 15 - m^2 = 0 \]
which simplifies to:
\[ (n + 10)^2 - m^2 - 85 = 0 \]
or:
\[ (n + 10)^2 - m^2 = 85 \]

### Step 4: Factor the equation
This can be factored using the difference of squares:
\[ (n + 10 - m)(n + 10 + m) = 85 \]

### Step 5: Find factor pairs of 85
The integer 85 can be factored into pairs of integers:
- \( (1, 85) \)
- \( (5, 17) \)
- \( (-1, -85) \)
- \( (-5, -17) \)

### Step 6: Set up equations for each factor pair
For each factor pair \( (a, b) \):
1. \( n + 10 - m = a \)
2. \( n + 10 + m = b \)

Adding these two equations gives:
\[ 2(n + 10) = a + b \implies n + 10 = \frac{a + b}{2} \implies n = \frac{a + b}{2} - 10 \]

Subtracting the first equation from the second gives:
\[ 2m = b - a \implies m = \frac{b - a}{2} \]

### Step 7: Solve for \( n \) using each factor pair
1. For \( (1, 85) \):
- \( n + 10 = \frac{1 + 85}{2} = 43 \)
- \( n = 43 - 10 = 33 \)

2. For \( (5, 17) \):
- \( n + 10 = \frac{5 + 17}{2} = 11 \)
- \( n = 11 - 10 = 1 \)

3. For \( (-1, -85) \):
- \( n + 10 = \frac{-1 - 85}{2} = -43 \)
- \( n = -43 - 10 = -53 \)

4. For \( (-5, -17) \):
- \( n + 10 = \frac{-5 - 17}{2} = -11 \)
- \( n = -11 - 10 = -21 \)

### Step 8: Collect all possible values of \( n \)
The possible values of \( n \) are:
- \( 33 \)
- \( 1 \)
- \( -53 \)
- \( -21 \)

### Step 9: Calculate the sum of all integers \( n \)
Now, we find the sum:
\[ 33 + 1 - 53 - 21 = -40 \]

### Final Answer
The sum of all integers \( n \) for which \( n^2 + 20n + 15 \) is a perfect square is:
\[ \boxed{-40} \]