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
Topic
Number Theory
Difficulty Level
Medium
Suggested Book
An Excursion in Mathematics

Start with hints

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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