I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these lecture notes.

For a computer algebra implementation, you could try Sage. I tried it myself, and it cannot find any nontrivial factorization of your polynomial.

I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these lecture notes.

For a computer algebra implementation, you could try Sage. But you will not find any nontrivial factorization of your polynomial for arbitrary complex coefficients $u_n$.

I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these lecture notes.

I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these lecture notes.

For a computer algebra implementation, you could try Sage. I tried it myself, and it cannot find any nontrivial factorization of your polynomial.