Showing that the result of an equation may be a perfect square

Question

Consider an equation that, for a few select integers $x$, yields a value that is a perfect square. In the case of perfect square quadratics it is easy. For example $(x-2)^2$ will always yield a perfect square value. But what about for an equation like $(x-2)(x-3)$? Or $x^4 -x^3 + x^2 + 2x + 1$? Perhaps it yields a perfect square for a few inputs, but is there a way to determine all such inputs?