Hi cBuG. Welcome to the world of digital photography!
It is very unusual to get only 4-6 shots on batteries, even using the LCD which consumes a lot of power. The HP Photosmart 812 digicam is known for having a good battery life.
My first question is what type of batteries are you using? Rechargeable NiMH AA batteries are recommended for your digicam.
If you use standard alkaline batteries, they are not the best choice (except in times when you have no other battery options) and will deplete quickly. You may be able to use AA Litihium batteries which are made for electronic devices such as digital cameras. They last a long time however their main drawback is that they are currently quite expensive.
If you are using NiMH batteries, realize that they can lose a small amount of charge even when not being used. So if you use your camera infrequently, charge the batteries a couple of hours before taking photos.
If none of these suggestions work, call HP. Your camera may be defective, or there may be a firmware upgrade available to address the problem.
If you find a solution, which I hope you do, let us know. It will help others if they run into a similar situation.
Related reading:Digital camera batterieshttp://www.digicamhelp.com/digital-camera-batteries/digital-camera-batteries.htmWhat is a firmware upgrade?http://www.digicamhelp.com/what-is-firmware-for-a-digital-camera/index.htm