You are simply better buying a new laptop since a new motherboard with a faster AMD E-series CPU will likely be expensive in comparison to the cost of a new laptop. That is if HP offered a more powerful AMD CPU for that specific laptop.
I recommend you buy one of the following laptops; both of which cost $379. The Acer is powered by the Intel Core i3-3110 (2.4GHz) and the Lenovo is powered by the Intel Core i303120 (2.5GHz). Intel CPUs are more powerful than all AMD CPUs, especially the AMD E-series, at the same clockspeed. If the Core i3 CPUs were clocked at 1.3GHz (same as E-300), it would still be able to run rings around the E-300.
15.6" Acer
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/www.newegg.com/Product/Product.aspx?Item=N82E16834314007
15.6" Lenovo...