Let x, y and z be the dimensions of the box.


Then the top and the bottom both (each) have the area xy = 252 in^2;

     the front and the back both (each) have the area yz = 105 in^2;

     and the two other opposite side faces have the area  xz =  = 60 in^2, each.


NOW

    the square of the volume of the box is


        V^2 = (xy)*(yz)*(xz) = 252*105*60 = 1587600.


    Hence, the volume itself is the square root from this value  V =  = 1260 cubic inches.

As you see, the solution is entirely build on using this trick  

    +--------------------------------------------------------------------------+
    |                                                                          |
    |  the square of the volume = the product of the areas of the three faces. |
    |                                                                          |
    +--------------------------------------------------------------------------+


If you know the trick (as you learned it now, after my explanations),

you will be able to easily solve any similar problem.