Can $n^2+4n$ be a perfect square?
Solution 1:
Clearly the number is greater then $n^2$, so the smallest square it could be is $(n+1)^2 = n^2 + 2n + 1 \neq n^2 + 4n$ because $2n +1\neq 4n $ for $n \in \mathbb N$.
The next square is $(n+2)^2 = n^2 + 4n + 4$ which is bigger than our number.
So in conclusion no, $n^2 + 4n$ is never a perfect square
Solution 2:
HINT: Recall that
$$(n+2)^2=n^2+4n+4$$