Prove that for no integer n, n6+3n5_5n4_15n3+4n2+12n+3 is a perfect square.

Hello Student
Please fidn the soln

n6 +3n5_5n4_15n3+4n2+12n+3 =
= n⁵(n+3) –5n³(n+3)+4n(n+3) + 3
=(n+3)[n⁵ –5n³+4n] + 3
=n(n+3)[n⁴-5n²+4] + 3
=n(n+3)(n²-4)(n²-1) + 3
=(n-2)(n-1)n(n+1)(n+2)(n+3) + 3

So this can not be perfect square

Thansk

Approved