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

Sn = n^2+20n+12,
n is a positive integer.
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.

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