Can i use my jensen jxp340 amp with my one 800watt 12" 4 ohm subwoofer?
Yes, technically you can, however you will be putting both your sub and your amp at risk. One of the worst things you can do to a sub is underpower it. The easiest way to explain it is n ot giving enough power to the sub leaves it starving for power. The motor structure ends up working too hard to try and move the cone properly and what happens is the sub will usually seize up rather than blowing out in the term you usually hear (ie, I had my system cranked up and I blew my subwoofer).
Secondly, in an attempt to get any real sound out of the under powered sub, you'll probably end up cranking the gain and bass boost on the amp which ends up overworking the amplifier quite a bit and can in turn lead to fried out amp chips, burnt up circuit boards etc.
Thirdly, by maxing the gain bass boost you're going to end sending a lot of distortion through your subwoofer in an attempt to get some decent bass. Distortion kills both subwoofers and speakers alike.
So to answer your question, yes you can hook up your subwoofer to that amp and yes, it will probably play for a while but I wouldn't recommend it. Look for a class d or mono block amp to push your sub as these will give you the most power and best heat dissipation. They are designed to push subwoofers specifically and you will get better sound and longer life out of your equipment. A good rule of thumb is look for an amp within 100-150 watts of both your RMS power handling and your Peak power. Remember, when it doubt, always go a little bigger on the amp side of the spectrum. It's easy to "tune down" an amp by having the gain and bass boost at a lower level (50%, 25% etc), than to turn 'em all the way up just to get some bass and end up prematurely blowing out your sub, amp or even worse, both.