find integral values of x such that value of x^2+19x+92 is a perfect square of an integer

We can just say

x^2+19x+92 = n^2,

where n is an integer.

Now just use the quadratic equation to solve for x:

x=\frac{-19 \pm \sqrt{19^2-4(92-n^2)}}{2}

Plug in whatever values of n you want, and you'll have your possible values of x.