Upgrade Sony VPCEA45FG to 8 GB RAM

Yes. Suggest you use a memory compatibility site and ensure the RAM is per this solution post:

The problem is not exactly the speed 1600Mhz memories ... also operate at 1333Mhz or 1066Mhz if the chipset can not run overclocked.
The problem is that the HM55 chipset does not recognize modules assembled with inserts RAM 512Mx8bits ... having your limit 256Mx8bits.
So make sure that the memory module that will buy your mount has taken 16 tablets 256Mx8bits, otherwise will not work.

https://communities.intel.com/message/233094#233094

Crucial's suggested RAM - http://uk.crucial.com/gbr/en/vaio-vpcea45fg/CT8230624#productDetails

Kingston (only single DIMMs listed) - https://www.kingstonmemoryshop.co.uk/kingston-4gb-4gb-x1-ram-memory-non-ecc-sodimm-1066mhz-pc3-8500-1-5v