Factorization of a complex polynomial

[hoopsmax25]

Homework Statement

p(x)=((x−1)^2 −2)^2 +3. From here find the full factorization of p(x) into the product of first order terms and identify all the
complex roots.

Homework Equations

I am having trouble doing this by hand. I know there are four complex roots but can't seem to figure out how to get them factored out.

The Attempt at a Solution

 

[SammyS]

Homework Statement

p(x)=((x−1)^2 −2)^2 +3. From here find the full factorization of p(x) into the product of first order terms and identify all the
complex roots.

Homework Equations

I am having trouble doing this by hand. I know there are four complex roots but can't seem to figure out how to get them factored out.

The Attempt at a Solution

Solve $\displaystyle ((x−1)^2 −2)^2 +3 = 0$ in steps.

First solve for $\displaystyle ((x−1)^2 −2)\,.$ There are two solutions.

Then isolate $\displaystyle (x−1)^2$ for each of the above solutions and take the square root of both sides in each case.