We bought a Samsung Soundbar x810 but the Subwoofer does not seem to be working properly although the blue and red led light up. I have tried rebooting/resetting per user manual instructions but it still does not work. Can someone please guess what the problem is and how I may solve it?

The sound bar and the subwoofer could of had quit working and and no longer works anymore. You will have to by a new soundbar and see if it can work.

Connect the power cords for the soundbar and the subwoofer to the back of each, and then plug both into a wall outlet. Use a small pointed object to press the ID SET button on the back of the subwoofer. Continue holding until STANDBY turns off and LINK (Blue LED) blinks quickly.

Pliug both soundbar and subwoofer in, Press and hold the "input" button on the soundbar for 4 seconds. The display should show two asterisks flashing. Press and hold the "pairing" button on the back of the subwoofer. The subwoofer light should turn blue (from red).

8 years late.

Because your TV is not Bluetooth-enabled and you are trying to connect it to a Bluetooth-enabled soundbar via Bluetooth. There are other ways to connect your TV to the soundbar:

1. analog AUDIO OUT (L/R) of your TV to the AUX IN on the soundbar (use a 3.5 mm.-to-RCA audio cable).
2. Optical OUT on your TV to Optical IN on the soundbar.
3. HDMI port to HDMI port (this may not work if your TV is not ARC compliant you have to try).

You get better sound quality with option 2.or 3. (if this one works for your TV)

Here is the link to download the soundbar's User Manual for help with the connections:

http://downloadcenter.samsung.com/content/UM/201411/20141114104022428/HW-H450-ZA-1114_Bookmark.pdf

I presume you need link the Wireless Subwoofer to tour Soundbar.
Here are instructions:
How To Link The Wireless Subwoofer To Your Soundbar HW Models Samsung

turn the soundbar off, then press and hold the reset button on the back of the subwoofer until the blue light starts blinking fast. the point the remote at the soundbar and hold the mute button until"ID SET" shows on the display. Then turn on the soundbar and it should find the sub and pair with it.

Check the receiver in the speaker and the amplifier. they both have to work to get sound.

Push in and hold the switch on the back of the subwoofer until the light starts blinking fast. Then, with the soundbar power turned off, hold the mute button on the remote. "ID SET" should show up on the display. Then turn on the soundbar, and the subwoofer should link to it.

May be thermal protection issue on component?? check..