My qustion is for the following

the problem reads as follows

(x-1)^2 = 7
x-1=+- sqrt 7
x= 1+- sqrt 7

the method they use was factoring in order to simplify the equation to make it easier is this correc?

x^2-9x-4=6
x^2-9x-10=0
(x-10)(x+1)=0

What method did they use here I think its simplifying but i am not sure?

4x^2-8x+3=5
4x^2-8x-2=0
2(2x^2-4x-1)=0
2x^2-4x-1=0

here the method that they use was first factoring then simplefiying and then they will use the quadratic formula is this correct?

You are right on all counts.