You really need to ask HP for a definitive answer, but I'll go so far as to say it's not likely this would work, and if it did, you wouldn't likely see much performance benefit. The reason is that the motherboard and RAM are customized for the processor your computer had when new. A 4 year old motherboard isn't likely to support all the features of a new CPU, but even if it somehow worked, the bus speeds and memory speeds wouldn't be likely to keep up with the increased speed of the I7 processor.
I have a 4 year old HP Pavilion dv4-2165dx that has an Intel Core i3 processor. Assuming I could find a processor with the right socket type, is it possible to swap the i3 for an i7? Would it be worth it/would I see a noticeable difference?