Algebra calculating to zero point without using rational root theorem

Question

I have to find the points where x equals zero in the following equation, without using rational root theorem.

The equation is: (3-x)(1-x)²+(1-x) = 0

I know the answer is x=2 and x=1. I get the x=1, that is simple.

Answer 1

$(3-x)(1-x)²+(1-x)$

$=[1-x][(3-x)(1-x)+1]$

$=(1-x)(4-4x+x^2)$

$=(1-x)(x-2)^2$.

I hope you can solve it from here.

Answer 2

$$(1-x)[(3-x)(1-x)+1]=(1-x)(3-4x+x^2+1)=(1-x)(2-x)^2$$