hi hwoarang,
I think this is the animated gif you are looking for:

Export["e:\\folder\\animatedgif.gif",
 Table[
  GraphicsGrid[{{Plot[x^2 + q^2 x, {x, 0, 10}, ImageSize -> size], 
     Plot[(x + q)^2 + 2 x, {x, 0, 10}, ImageSize -> size]}}
   ],
  {q, 5}],
 "DisplayDurations" -> 0.5,
 Background -> LightBlue]

DisplayDurations is 1 picture each 0.5 second. In total 5 pictures by using Table.

Thank you,Pieter for the reply.

I agree that "Table" works well, and I did with this. But the interesting problem is that when one tries to use Plot function directly with Animate, the created movie file does not work. i) GIF file is not moving. ii) Movie files (swf or avi) are always moving both direction - forward and backward even though I have tried all options that I could do (including "AnimationDirection"). And also I have found that there were some persons who faced same problems in archive. But there was no one who had the solution for this. Do you have some ideas ?

Have a nice day. Thank you.

Hwoarang.