UK Trademarking

Hey all

Hope everyone is well!

I am after some advice regarding trademarking in the UK, which is where I am based.

If registering your ltd company with a relatively nondescript name for example Smith Enterprises Ltd with the intent to use a trading name/brand name such as Bananafy or Orangely (completely hypothetical names) would/could you trademark both the registered company name itself as well as the trading name(s).

Also how would this work in the situation whereby you have a registered company name such as Facebook Corp or for the purposes of this UK example – Facebook Ltd but your actual product/brand name is Facebook, would you just trademark the registered company name or would you trademark both the registered company name as well as the product/brand name Facebook – i.e. would just trademarking the registered company name Facebook Ltd protect your use of the brand name 'Facebook'.

Are trading names/brand names actually able to be trademarked or is it just the registered company name which can be trademarked?