First part = x
Second part = x+d
Third part = x+2d

The two equations are

x + (x+d) + (x+2d) = 24
      x(x+d)(x+2d) = 440

Simplifying the first equation:

x + (x+d) + (x+2d) = 24
x + x + d + x + 2d = 24
           3x + 3d = 24
             x + d = 8
                 d = 8-x

Substituting in the second equation:

      x(x+d)(x+2d) = 440
x(x+8-x)[x+2(8-x)] = 440
     x(8)[x+16-2x] = 440
          8x(16-x) = 440
           x(16-x) = 55
            16x-x² = 55
        -x²+16x-55 = 0
         x²-16x+55 = 0
       (x-11)(x-5) = 0

x-11 = 0 ;  x-5 = 0
   x = 11;   x = 5
d = 8-x  ; d = 8-x 
d = 8-11 ; d = 8-5
d = -3   ; d = 3

One solution:
First part = x = 11 
Second part = x+d = 11+(-3) = 8
Third part = x+2d = 11+2(-3) = 11-6 = 5

11, 8, and 5

Other solution:
First part = x = 5 
Second part = x+d = 5+3 = 8
Third part = x+2d = 5+2(3) = 5+6 = 11

5, 8, and 11

So they are the same. 

(The two solutions come from whether you chop 
a 24-inch stick starting from the right end 
or starting from the left end.)

Edwin