The difference between PCIe x1, x4, x8, x16 and x32

pcie 8x slot
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Yes; As long as the card will fit in your case, a PCIe x4 slot can handle the card you describe Having no other information I can only guess

PCI Express adapters use a different type of slot than Peripheral Component Interconnect and Peripheral Component Interconnect-X

pcie 4 gpu in pcie 3 slot Should work fine, it's the same as putting a 16x card in to a slot that's only wired for 8x or 4x, you don't get the full bandwidth but it will

pcie 4 slot Basic adapters, which simply run traces from each connector directly to corresponding lanes on the PCIe slot , #1 goes to the first