Final answer:
The question requires finding the completely factored form of the polynomial f(x) = x^3 + 4x^2 + 7x + 6, which involves finding its roots and factoring it completely.
Step-by-step explanation:
The completely factored form of f(x) = x^3 + 4x^2 + 7x + 6 can be found by looking for real numbers that, when substituted for x, will make the polynomial equal to zero. These values are the 'roots' or 'zeroes' of the polynomial and correspond to the factors of the polynomial. The Rational Root Theorem can assist in finding possible rational roots, which are among the factors of the constant term (6) over the factors of the leading coefficient (1) in this polynomial. Once we find a valid root, say a, we can factor out (x-a) from the polynomial. We then continue factoring the resulting quadratic using techniques like factoring by grouping, the quadratic formula, or completing the square until the original polynomial is fully factored.