$\frac{14n+25}{2n+1}=k^2\Longrightarrow 7+\frac{18}{2n+1}=k^2$
$\Longrightarrow 2n+1\in D_{18}=-18, -9, -6, -3,-2, -1,~1,~2,~3,~6,~9,~18 ~~$ (divisors of $18$)
$\Longrightarrow n\in \{-5,~-2,~-1,~0,~1,~4\}$
of which only $n=\boxed{-2,~0,~4}~$ make the given fraction a perfect square.