Is it a perfect square?

Algebra Level 4

Determine the number of integers \(n\) for which \[{ n }^{ 6 }+{ 3n }^{ 5 }-{ 5n }^{ 4 }-{ 15n }^{ 3 }+{ 4n }^{ 2 }+{ 12 }n+3\] is a perfect square.

(This problem is a modified form of the one that appeared in KVS JMO 2014)

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