Technically, you should be able to -- the manual says it works with the quad-core i5 & i7 CPUs (http://ftp.dell.com/Manuals/all-products/esuprt_desktop/esuprt_inspiron_desktop/inspiron-3847-desktop_Reference%20Guide_en-us.pdf), but it's a "generic" notation (i.e. no specific models are listed).
As far as which CPUs specifically work, this model (https://www.cnet.com/products/dell-inspiron-3847-core-i3-4130-3-4-ghz-4-gb-1-tb-english/specs/) apparently had the i3-4130, this model (https://www.cnet.com/products/dell-inspiron-3847-core-i5-4440-3-1-ghz-8-gb-1-tb/specs/) had the i5-4440, & this guy (http://www.tomshardware.com/answers/id-2597809/change-cpu-dell-inspiron-3847.html) had the "Haswell Refresh" i3-4150.
Some further digging turned up this guy's build (https://pcpartpicker.com/b/RBtJ7P) -- it appears that he has the older Lynx Point (H81 chipset) motherboard, but it came with the Haswell Refresh i5-4460. So, depending on your BIOS, you will probably have little to no trouble installing just about any Haswell CPU in there. My strong suggestion, however, would be to avoid any unlocked "K" CPUs. For one, I have yet to see an OEM board that allows for overclocking (not counting the special "gamer's edition" PCs that come with overclockable hardware & cooling systems pre-installed). For another, you'll not only pay more for the CPU, you'll have to then buy a CPU cooler.
Now, the question is whether it's worth upgrading the CPU in the first place. If you already have a Core i5 in this system (or especially a Core i7), then forget about it; all of the Haswell i5/i7 CPUs are still 1st-tier (http://www.tomshardware.com/reviews/cpu-hierarchy,4312.html), so unless you need an i7 for more streaming threads it's not going to do much. If you have an i3, you're in better luck, as those are 3rd-tier CPUs; going to an i5 or i7 would be a decent boost. And if you're "rocking" a dual-core Pentium...oh yeah, it's time for a new CPU to replace that 6th-tier junk...
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