This is the 451 underseat bass box, it is available from smart (part A4518200302 for RHD)
for a price of 125 at the time of writing, although smart part prices change almost daily.

A better idea is to buy one second hand from someone who is upgrading.

Remember to get the correct one for your car depending if you have an RHD or LHD smart.

This shows the underside, you can see the speaker and the sound port.

Remove the driver's side seat, run the wiring and place the bass box in position.

Refit the seat, this holds the bass box in position.

Plug the wiring connector into the bass box.

Now you have to wire the bass box to the stereo. The bass box connections are numbered.

Here are the connection details for the rear of the stereo.

Working with the connection numbers and wiring colours, the correct wiring is as follows.

Bass Box Pin Number Wire Colour Stereo Connection
1 Brown Green B6
2 Blue Brown B2
3 Red Blue A4
4 Green B5
5 Blue B1
6 N/A No Connection
7 Brown Car Body Earth
8 Black A5

Note that bass box pin 7 doesn't go to the stereo, it should be attached to an earth point.
This can be any point you want, the easiest being one of the seat bolts.

Splice each of the wires into place or use straight crimp connections. For a less permanent wiring job you could buy a short ISO extension cable, splice into that and make it plug and play.

This means you would have to cut any of the original wiring behind your stereo.

What's Inside The Bass Box

This is very much like a bag of crisps, a lot of air and not much contents.

The speaker is a 13cm dual voice coil subwoofer with a 58mm depth.
If you wanted to replace it you could cut the apperture out and fit a larger diameter speaker
but the maximum mounting depth of the new speaker is going to be about 60mm.

The amp circuit has no info on it but I suspect it is probably about 25 watts.

Looks Rubbish, Can I Make My Own?

You aren't wrong there, it is made of plastic thinner than that used on the 450 under dash boxes.
There is still an average amount of room under the seat of a RHD 451. Check your LHD ones yourself as you may have the extra height restriction with the lower seat and extra electics.

Below shows the usable space on a RHD 451.

The 80mm maximum height at the back of the seat reduces the number of
aftermarket bass boxes that you could fit under there but any of these will be an improvement over the standard sound.