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# Understand the problem

find all integers $x$ such us $1+x+{x}^{2}+{x}^{3}+{x}^{4}$ is a perfect square.

##### Source of the problem
New Zealand team training 2004
Number Theory
Medium
##### Suggested Book
An Excursion in Mathematics

Do you really need a hint? Try it first!

Try to show that the given expression (or a suitable multiple of it) lies between two consecutive squares.

Depending on the value of $x$, there are two possibilities: $(8x^2+4x+3)^2<64(1+x+x^2+x^3+x^4)<(8x^22+4x+4)^2$ and $(8x^2+4x+2)^2<64(1+x+x^2+x^3+x^4)<(8x^22+4x+3)^2$. Find out the conditions under which each inequality is valid.
Show that, for $-1\le x\le 3$, none of the inequalities in the previous hint hold.
Verify that the only solutions are $x=-1,0,3$.

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