TITLE: Find all integer solutions: $x^4+x^3+x^2+x=y^2$
QUESTION [5 upvotes]: Find all integer solutions of the following equation:
$$x^4+x^3+x^2+x=y^2$$

REPLY [12 votes]: When $|x|>1$ we have 
$$(2x^2+x)^2<4(x+x^2+x^3+x^4)<(2x^2+x+1)^2.$$