Final answer:
The complete factorization of the polynomial x^3 + 2x^2 + x + 2 involves finding the roots and corresponding factors starting with (x+1) and then using either factorization or the quadratic formula for the resulting quadratic polynomial.
Step-by-step explanation:
The complete factorization of the polynomial x^3 + 2x^2 + x + 2 can be found by looking for a common factor or by using factorization methods such as synthetic division, grouping, or trial and error with possible rational roots. However, as this polynomial does not have an obvious common factor or easily groupable terms, we have to try other methods. By applying the Rational Root Theorem, we can find that one of the roots of the polynomial is x=-1. Once we find one root, we can divide the polynomial by the binomial corresponding to that root and simplify it further to find the other factors.
Firstly, we confirm if x=-1 is a root by substituting into the polynomial which gives: ((-1)^3 + 2(-1)^2 + (-1) + 2 = 0). Since it equals 0, -1 is a root, and (x+1) is a factor. Next, we perform the division of the original polynomial by (x+1) which results in a quadratic polynomial, that we can then factor or solve using the quadratic formula if necessary to find the remaining factors.