<s>[INST] What are integers $x$ and $n$ such as $ x^4+x^3+x^2+x+1 = n^2$? What are the integer numbers $x$ such as $x^4+x^3+x^2+x+1$ is the square of an integer?
$0$ and $-1$ are obvious answers, but how many are there?
 [/INST] Idea: Bound between known perfect squares.
For $ x \neq 0$, we have (expand and verify)
$$ ( x^2 + \frac{x}{2} ) ^ 2 < x^4 + x^3 + x^2 + x + 1 < (x^2 + \frac{x}{2} + 1)^2. $$
Hence, the only possibility for a perfect square is for $x$ to be odd, and the square is equal to $ ( x^2 + \frac{x}{2} + \frac{1}{2})^2$. This gives us (expand and simplify) $ \frac{1}{4} (-x^2 + 2x + 3) =0$, hence $x=-1, 3$.
Adding in $x=0$, we get 3 solutions.
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