You are given


    x + 125 = n^2     (1)

    X + 176 = m^2     (2)


Subtract (1) from (2).  You will get

    m^2 - n^2 = 176 - 125,   or

    m^2 - n^2 = 51.


It implies

    (m+n)*(m-n) = 51.     (*)


          It is my basic equation, and I am going to find all its integer solutions.

          Notice, that if the pair of positive numbers (m,n) is the solution, then 3 other pairs 
          (-m,-n), (-m,n) and (m,-n) are the solutions, too.

          Therefore, I will look and search for positive solution pairs (m,n) only, keeping in mind that 
          every such a pair brings 3 other solutions, playing with their signs.

          But since my "x" depends only on m^2 and n^2, this playing with signs does not matter, at all.


Since "m" and "n" are integer numbers, it implies that 

EITHER

        m + n = 51,     (3)
        m - n =  1,     (4)


OR

        m + n = 17,     (5)
        m - n =  3.     (6)


From equations (3), (4), the solution is  m= 26, n= 25.

From equations (5), (6), the solution is  m= 10, n=  7.


Thus EITHER

        x + 125 = n^2 = 25^2 = 625, which implies  x= 625-125 = 500,


     OR

        x + 125 = n^2 = 7^2 = 49,  which implies  x= 49-125 = -76.


ANSWER.  The problem has two solutions .  The number under the question is EITHER 500  OR  -76.