Math Problem: When is Sn = n^2+20n+12 a perfect square?

This is a question that appeared in RMO in 2012 I guess.

Let 

1. Sn = n^2+20n+12,  
2. n is a positive integer. 
3. What is the sum of all possible values of n for which Sn is a perfect square.

 Solution:

 

The solution was suggested by one of my colleagues Ashish Panda.