At newegg.com, the best new laptop you can buy is the following Acer V5-552-8404 AMD A-Series A8-5557M(2.10GHz) 4GB for $420. The quad core APU has a Radeon HD 8550g graphics core which is simply the most power graphics solution within your budget. The laptop comes with 4GB of embedded (soldered in) RAM with a free slot for up to an 8GB stick of RAM.
The only problem is that with just one stick of RAM the RAM will be operating in single channel mode instead of dual channel mode. That means your performance will not be the best unless you install another stick of RAM.
http/www.newegg.com/Product/Product.aspx?Item=N82E16834314228
According to Crucial's website, the following 4GB stick of RAM for $48 is guaranteed to work with the laptop. Crucial is a major name brand so it costs a few extra $$ than lesser name brands.
http/www.newegg.com/Product/Product.aspx?Item=N82E16820148736
http
/www.crucial.com/store/mpartspecs.aspx?mtbpoid=0AFE03D1A5CA7304
The following G.Skill 4GB RAM stick is $40, but not 100% guaranteed to work with the laptop. Chances are it will though.
http/www.newegg.com/Product/Product.aspx?Item=N82E16820231695