the method is as follows.
first factorize:
x^2 + 19x + 88
= (x+8)(x+11)

One solution is straightaway that either x = -8 or x = -11 as the product will become 0 which is a perfect square.

Second, for it to be a perfect square other than 0, the 2 numbers must have a diff. of 3 and their product must be a perfect square.
write numbers from 1 to 10 , and write the product formed by them and numbers 3 more than them.
1,4 = 4 ... 2,5 = 10 ....3,6 = 18....4,7 = 28.... upto 10,13 = 130
thus we see that the only case is 1,4 = 4
thus x+8 = 1 and x+11 = 4 gives x = -7
and x+8 = 4 and x+11 = 1 gives x = -12

Thus the solutions are x = {-7, -8, -11, -12}

 

*PLZ RATE*