The top is a 10x10 square, so 100 little cubes get painted.
The bottom is a 10x10 square, so 100 little cubes get painted.
That's 200 so far...

Now, paint the front and back of the outside...

The front is a 10x10 square...  that's 100 little cubes...  But, we've already counted the top stripe of cubes and the bottom strips since these got hit when we did the top and bottom.  So, for the new little cubes we're hitting, we have an 8x10 grid...  that's 80 more little cubes getting painted.

Same for the back... another 80.

So far:  100 + 100 + 80 + 80

Now, we do the left and right side of the big cube...  Thinking about it the same way we did above, we'll be hitting an 8x8 set of new cubes with paint...  That's 64 for the right side and 64 for the left side.

Total: 

100 + 100 + 80 + 80 + 64 + 64
= 488