Find all integers $x$ such that $x^{4}+x^{3}+x^{2}+x+1$ is a perfect square. (Bounding)
Let $A=x^{4}+x^{3}+x^{2}+x+1$. We have $x^{2}-x+1>0$ (quadratic with a strictly negative discriminant) and by immediate induction, for $x \geqslant 2, x^{3}>x+1$. We then have: $\left(x^{2}+x\right)^{2}=x^{4}+2 x^{3}+x^{2}>$ $A>x^{4}+2 x^{2}+1=\left(x^{2}+1\right)^{2}$. Therefore, there is no solution for $x \geqslant 2$. We notice that $x=1$ does not work and that $x=0$ works.