Final answer:
The partial fraction decomposition of the given function, (6x^2 + 10x + 28) / (x^3 + 3x^2 + 4x + 12), involves finding the functions f(x) and g(x) which are the numerators that correspond to the decomposition terms. The correct forms of f(x) or g(x) are not provided in the question, and further steps like polynomial division would be required to find them.
The correct answer is b) (x + 2)
Step-by-step explanation:
The partial fraction decomposition of a rational function involves expressing the function as a sum of simpler fractions, where the denominator of each term is a factor of the original denominator. For the given function (6x^2 + 10x + 28) / (x^3 + 3x^2 + 4x + 12), one way to begin the decomposition is by dividing the numerator by the denominator, if possible.
However, since there is no obvious way to simplify this directly, we must approach the problem by considering the possible factors of the denominator and then finding the numerators, i.e., the functions f(x) and g(x), that would correspond to each decomposition term.
In partial fraction decomposition, f(x) and g(x) are not the original numerator but are found such that when you add the decomposed fractions together and multiply by the common denominator, you should get back the original numerator.
Thus f(x) and g(x) would represent the numerators corresponding to the factors of the denominator. Answer options (a) through (d) give potential forms for these functions, and we would need additional work, such as polynomial long division and equating coefficients, to determine which is correct.
The correct answer is b) (x + 2)