Question

how do you factor the polynomial 3x^4-4x^3+4x^2-4x+1?

i got (x-1)(3x-1)(3x^2+3)

but the answer is (x-1)(3x-1)(x+i)(x-i)

Answer 1

You should have realized your answer was wrong by looking at the constants of your three factors.
Multiplying them would have given you a last term of +3 instead of the needed +1.

You were close, actually you are off by only a constant factor of 3

Look at your last factor of (3x^2+3)
=3(x^2+1)
=3(x+i)(x-i)

Compare that with the actual answer given.
Looks like you just made an arthmetic error somewhere in your division.

Answer 2

3x^2+3 = 3 x(x^2 + 1) = 3(x+i)(x-i)

You only got an overall factor of 3 wrong.

Answer 3

where does the "3" go then?

3x^2+3 = 3 x(x^2 + 1) =

"3"(x+i)(x-i)

because the answer is:

(x-1)(3x-1)(x+i)(x-i)