<s>[INST] Find the real ordered pairs $(x,y)$ satisfying $x^2+x=y^4+y^3+y^2+y.$ Find the real ordered pairs $(x,y)$ satisfying $x^2+x=y^4+y^3+y^2+y.$

$x^2+x=y^4+y^3+y^2+y$
$4x^2+4x=4y^4+4y^3+4y^2+4y$
$4x^2+4x+1=4y^4+4y^3+4y^2+4y+1$
$(2x+1)^2=4y^4+4y^3+4y^2+4y+1$
I am stuck here.Is there a general method to solve such type of equations?Please help me.
 [/INST] Under the assumption that the problem asks for integer solutions, (which is probably the case) 
We have $$(2y^2+y)^2<(2x+1)^2=4y^4+4y^3+4y^2+4y+4<(2y^2+y+1)^2$$ for $y > 3$ or $y  <-1$, so we must have $-1 \le y \le 3$. Now bash.
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