polynomial equation solve in full numbers

Question

$x^2+x+41=y^2$ <--solve that in full numbers. I get to that point $(y-sq(x))(y+sq(x)))=x+41$ which imo must be false because this implies that $sq(x)^2$ is equal to $-x$ did I make a mistake somewhere Ok I can't solve it, I'll have some help

Answer 1

You can multiply your original equation by $4$ to obtain $$4x^2+4x+164=4y^2$$

Then complete the square on the left $$(2x+1)^2+163=4y^2$$

Now rearrange to obtain the difference of two squares $$(2y)^2-(2x+1)^2=163$$Factorise the difference of two squares $$(2y+2x+1)(2y-2x-1)=163$$

If this is to be solved in integers, note that $163=163\times 1=-163\times -1$

That should be enough to get you started.