Final answer:
The correct partial fraction decomposition of 7x² + 14 / (x² + 3)² is 7/(x² + 3) - 7/(x² + 3)², that is, option 1. This decomposition follows the structure expected for a rational expression with a perfect square in the denominator.
Step-by-step explanation:
The partial fraction decomposition of 7x² + 14 / (x² + 3)² involves expressing the rational expression as a sum of simpler fractions whose denominators are factors of the original denominator. To do this, we compare the given options with the structure we expect for partial fractions of this kind. The denominator (x² + 3)² is already a perfect square, and since our numerator is a polynomial of a degree less than the denominator's, we do not expect terms with x in the numerators of the partial fractions. Therefore, we expect terms of the form A/(x² + 3) and B/(x² + 3)², where A and B are constants to be determined.
By the structure, we see that the correct partial fraction decomposition must balance the degrees of the polynomials in the numerator and denominator when summed. Therefore, option 1, which states 7/(x² + 3) - 7/(x² + 3)², fits the expected form and is the correct decomposition after ensuring that the numerators sum up to 7x² + 14 when combined over a common denominator.