Ran away from it, don't walk, run.
PE is print receptive for a while after being molded/extruded/cast what-have-you, but shortly thereafter becomes very tough to get decent adhesion on, if you insist, it will likely need to be "primed" to get decent ink adhesion, this can be done in 3 different ways.
1 - Corona discharge. Similar to a big electrostatic shock, most automated printing presses will have this in-line with the print process. This method of priming lasts for a while, but not for-ever, after sitting for a while the material will lose the print-receptive properties that corona discharge gave it.
2 - Propane Flame treating, letting a propane flame "lick" over it will give it temporary print-receptive properties. when done properly it looks like you've fogged it with breath and then the humidity evaporates very quickly as the flame moves away. this isn't what's happening, but that's how it looks.
3 - Chemical pretreatment. nasty, solvent based primers are wiped, sprayed or dipped on and allowed to dry and act as an adhesive agent between the ink and PE. you can get spray cans of the primer at autobody supply shops as many palstic bumpers and body parts require the priming to get adequate paint adhesion.
Bags like you posted are usually web-processed with corona discharge applied seconds before printing, sometimes post-treated too in highly specialized machines meant just to print on such a difficult substrate.
You'll need a solvent based ink, or UV-cure ink meant to adhere to one of the three methods of priming, and once you are all done, the print will likely be "pretty-good" as far as adhesion, it will never truly "bite-in" to the plastic, just sit on top, sort of glued to the product, flexing, stretching or mechanical wear on the bag's print will probably show the short comings of your process unless you spend the considerable time to R&D a really top-notch process or buy a machine/process meant to do just this job in very high quantities very quickly.
I print almost anything in my shop, textiles, glass, metals, almost all plastics, rubber, silicone, ceramics, powder coating, anodizing and even rigid polyethylene and it's evil twin, polypropylene, but if someone wants me to print on PE bags (other than non-woven shopping bags) I politely decline.
Even if you gear-up and decide to print it, it's almost impossible to get paid enough to make it worth-while. The bags are just too cheap, and the print ends up costing many times what the bag is worth. Tough to sell even a $0.50 print onto a $0.08 bag.
That all said, that bag kinda looks like it may be an anti-static bag, or metalized PE.in which case the print may not be going onto PE, but onto the metalic surface put onto the bag to conduct static charges away from sensitive (and often pricey) electronics, in which case $1 a print may be easily absorbed into the products end-price, and you may find a traditional ink may adhere to the surface treatment of the bag. At my shop I have 6-7 possibly suitable different ink lines I just grab and do simple adhesion tests with, if you don't have that luxury, sending a sample of the bag to Nazdar to test is probably your best bet.
