How can I find integer values for which a given expression gives a perfect square? Find the integer values for which $x^2+19x+92$ is a perfect square.
Also, How to proceed if you have to find values ( not necessarily integer)?

Complete the square:
$$\begin{align}
x^2 + 19x + 92 &= m^2\\
\iff (2x)^2 + 2\cdot 19\cdot (2x) + 368 &= (2m)^2\\
\iff (2x+19)^2 + 7 &= (2m)^2\\
\iff (2m)^2 - (2x+19)^2 &= 7.
\end{align}$$
Since $7$ is prime, that means $2m = \pm 4$ and $2x+19 = \pm 3$.
