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

$$(n+2)^2=n^2+4n+4$$