Let z = one number
Let x = other number
Let y = their product = xz

So y = xz

Since their difference is 4

z - x = 4
    z = 4+x

   y = xz
   y = x(4+x)
   y = 4x+x²
   y = x²+4x

This is a parabola that opens upward:


   
So its lowest point is the minimum value of y.  That is the vertex.

The x-coordinate of the vertex is  =  = -2

Its y-coordinate is found by substituting x = -2 in

   y = x²+4x
   y = (-2)²+4(-2)
   y = 4-8
   y = -4

So the vertex is (-2,-4)



Since y = the product, the minimum product is -4.

Edwin