Number theory vs perfect square Everyone help me please! Find for all $x$ is positive Such that $x^2+5x+2$ is perfect square of integer First I suppose $x^2+5x+2= y^2$ and substract $y^2$ from both side Got $(x-y)(x+y)=-2x-5$ But I can't consider in case by case

Take $x^2+5x+2=y^2$ for some integer $y$. Then from the quadratic formula, we know that $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$ $$x=\dfrac{-5\pm\sqrt{5^2-4(2-y^2)}}{2}=\dfrac{-5\pm\sqrt{25-8+4y^2)}}{2}=\dfrac{-5\pm\sqrt{17+4y^2)}}{2}$$

Now, for $x$ to be an integer, $4y^2+17$ should be a perfect square. So, take $4y^2+17=k^2$ for some $k$.

$$4y^2-k^2=17$$ $$(2y+k)(2y-k)=17$$

Now we have four combinations of $(2y+k)$ and $(2y-k)$.

Can you take it from here?