Partial factorization?

[DrD]

factors(16*x^8+8*x^4+1); ==> [2*x^2-2*x+1,2,2*x^2+2*x+1,2]; Unfortunately, (4*x^4+1)^2, is not included in the results vector.

This is an important factorization. A problem solving process I was validating, led to finding the square root of 16*x^8+8*x^4+1. By hand, finding the square root of an expression squared, is easy. I was using the prime to confirm, step by step, a lengthy problem, where understanding the problem solving method is just as important as the solution to the problem.

Can you think of a way to factor 16*x^8+8*x^4+1, (using the prime), that does include (4*x^4+1)^2?
I needed it to find a square root, and Didier's approach gets there, but more in general, I haven't found a command that includes a full factorization chain.

Thanks,

-Dale-