Need Help, Helping a Friend Pick a Laptop

R500VM-MS71 15.6"
Intel® Core(tm) i7-3610QM
8GB DDR3-1600 SDRAM
750GB 5,400RPM Hard Drive
15.6" HD Widescreen Display
DL DVD±RW/CD-RW
Memory Card Reader
Nvidia GT 630M
10/100/1000 Network
802.11b/g/n Wireless
Microsoft Windows 7 Home Premium (64-bit)

Sony Vaio S 13.3"
3rd gen Intel® Core™ i5-3210M processor (2.50GHz / 3.10GHz with Turbo Boost)
Windows® 7 Home Premium 64-bit
13.3" LED backlit display (1366 x 768)
Intel® HD Graphics 4000 with Intel® Wireless Display technology
320GB (5400rpm) hard drive
4GB (4GB fixed onboard + 1 open slot) DDR3-1333Mhz
CD/DVD player / burner
Internal lithium polymer battery (4400mAh)

1. What is your budget?
$800 before tax at the moment

2. What is the size of the notebook that you are considering?
Anything from 13 inches to 16 inches.

3. What screen resolution do you want?
1080p is preffered of course but I do not know if its possible.

4. Do you need a portable or desktop replacement laptop?
Portable

5. How much battery life do you need?
At least decent life ~ 4 hrs at least, more heavily preferred.

6. Do you want to play games with your laptop? If so then please list the games that you want to with the settings that you want for these games. (Low,Medium or High)?
Essentially no gaming (other than flash/facebook haha)

7. What other tasks do you want to do with your laptop? (Photo/Video editing, Etc.)
Video streaming, surfing, other general college computing, and Mathematics software (mathmatica and stuff like that.) - she is a math major.

8. How much storage (Hard Drive capacity) do you need?
500gb+

9. If you are considering specific sites to buy from, please post their links.
Anywhere but Brick and Mortar is probably preferred, so Microcenter, Bestbuy, Walmart, Samsclub, etc.

10. How long do you want to keep your laptop?
Probably more than 4 years and want it to function well for those years

11. What kind of Optical drive do you need? DVD ROM/Writer,Bluray ROM/Writer,Etc ?
At least a DVD reader, but I guess up to a writer is typically standard

12. Please tell us about the brands that you prefer to buy from them and the brands that you don't like and explain the reasons.
Well Asus and Lenovo's Thinkpads are prefered. The Vaio's look nice but the E series doesn't look to great. HP and Acer appear to be plasticy and not as well built but really solid versions of these laptops are ok at least for suggestions

13. What country do you live in?
USA

14. Please tell us any additional information if needed.
She is going to be a college student, so at least reasonably portable, but that Asus above was considered to be a "pretty good size".