Cheap? Good IQ? Long zoom? Generally, I would say "Pick 2!"
BTW, x?? zoom means almost nothing unless the short end focal length is specified - a x10 zoom could be a 50-500mm or a 24-240 - for your needs/wants, the 50-500 would be ideal but for my tastes, a decent wide angle at the short end is far more important! At a guess, any pocketable camera will struggle to get to a long telephoto and even if it does, chances are that it will struggle to focus and will be rather slow (high f number).
My advice would be to resign yourself to spending a little more and buying from a real shop on the High Street. Take a card (SD seems to be fairly ubiquitous these days) into a well stocked shop (Jessops springs to mind - some of the staff in some of the shops are actually knowlegeable and helpful!) and try a few that meet the specs you're after. Only fair that you spend your money with the shop that lets you get your sticky paws on their merchandise rather than just use them as a showroom for the internet!