Pving parameters of perfect cubes and squares

[frosh14]

Show that there are no positive integers n for which $n^4 + 2n^3 + 2n^2 + 2n + 1$ is a perfect
square. Are there any positive integers $n$ for which $n^4 + n^3 + n^2 + n + 1$ is a perfect square?
If so, find all such $n$.

[Moon]

Hint:

Try to show that $n^4 + 2n^3 + 2n^2 + 2n + 1 $ lies between two consecutive squares.

[Tahmid Hasan]

1.$n^4<n^4 + 2n^3+2n^2+2n+1<(n+1)^4$
the second problem can be solved in the same way.

[nayel]

1.$n^4<n^4 + 2n^3+2n^2+2n+1<(n+1)^4$
the second problem can be solved in the same way.

There can be perfect squares between two consecutive fourth powers, e.g. $1^4<2^2<2^4$.

[Tahmid Hasan]

sorry for my miscalculation ,here's the right one
$(n^2+n)^2<n^4+2n^3+2n^2+2n+1<(n^2+n+1)^2$