Someone could explain to me with words, and in a extended way, this part of this

ID: 3564567 • Letter: S

Question

Someone could explain to me with words, and in a extended way, this part of this problem.

It is from "Chegg Guided Solutions for Discrete Mathematics 7th Edition Chapter 2.4 Problem 3E"

My doubts are,

In the part 2, how did they manage to form (k+1+1) from (k+1)[(k+1)!],

wouldn't be a k^2 or something? also I don't know how to work with negation signs in algebra, did they change something?

And in the last part, how did they get (k+2)! from (k+1)! (k+2) part?

Maybe I'm missing a rule or something, but please, explain to me in a clear way.

In part 2 they are taking [(k+1)!] as common since there are two terms containing (k+1)!
so after taking common we are getting (k+1)[(k+1)!]

now for (k+2)!

since (k+1)! = 1*2*3*4*......*k*(k+1)
so if we multiply (k+2) with (k+1)! we will get (k+2)!
this is a basic rule of factorial

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