$2x^3 + x + 8 = y^2$
WolframAlpha tells me that there are no integer solutions to this equation. But how could I deduce this from observation? If so, the next step, I guess, is proving that the $LHS$ can never be a perfect square.
The quadratic which appears in the post does have integer solutions (as pointed out in the comments) but the attached WA link refers to the equation $2x^3+x+8=y^2$ which indeed has no integer solutions.
To see that the cubic equation has no integer solutions we work $\pmod 3$. Since $n^3\equiv n\pmod 3$ for all $n$, we see that the LH is always $2\pmod 3$. But $2$ isn't a quadratic residue $\pmod 3$ so we are done.