I found this one, it right at your budget though! Its cosmetically nice and the specs are high. Of course your gonna get nerds on here that give you links to 4k ones and try flex theres or give you advice but say they added this and that, lmao. Inspiron 15 5000 Gaming Series
7th Generation Intel® Core™ i5-7300HQ Quad Core
Windows 10 Home 64bit English
8GB Single Channel DDR4 2400 Mhz
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Buy a cheap laptop for your lectures and a desktop PC for your gaming.
Spending £1,000 on a laptop would get you something good, but it'd get you a better desktop. Bottom line is, what games? What are the max specs for the games you're going to be playing? Is it worth spending more than necessary?
Just search around, I've noticed buying something branded is a total rip off. (ex. Alienware, Mac).
I'd personally get a small touch screen tablet or computer for the lectures. All of the information is online anyway, I personally didn't take many notes in my lectures. But hey ho if your budget is £650 - £800 I doubt you'd get a brand new good Apple Mac.
I'm not for sure if you like the brand Acer but my buddy has an Acer gaming laptop (I don't know the specific specs), but he get 120 fps on Fortnite while recording and is capable of taking notes with word etc.
If I find the laptop online I will send you the link!