Our potted palm has dripped or sprayed a very sticky substance all over everything near it: curtains, carpet, furniture, plastic case, etc. It is truly awfully sticky!
What is the best remover?
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Check the palm thoroughly, sounds as if its got an infestation of either mealybug, scale insect or a heavy crop of aphids. These all produce honeydew, and that is very sticky indeed, and may be what's 'sprayed' all over everything. If it is infested, spray thoroughly with a suitable insecticide, but you probably need to take it outside to spray it, then return it indoors once the spray has stopped dripping.
As for removing it from furnishings, wash anything that can be washed in the machine, and it sounds as if you'll need the carpet cleaned, at least in that area, though you could try Vanish carpet cleaner in a spray, intended for high use areas (if its available where you are), not sure whether it'll work though. Any hard surfaces such as the plastic case can be sprayed with a kitchen surface cleaner and washed in that way.