Final answer:
Partial fraction decomposition is used to express a complex rational expression as a sum of simpler fractions. For the given expression, it involves rewriting the numerator so that it has a degree less than the denominator's, and determining constants for the terms in the decomposition.
Step-by-step explanation:
The question pertains to the partial fraction decomposition of a given rational expression. The expression provided is 15x^3 + 25x^2 + 5x + 2 over (5x^2 + 1)^2. When approaching the problem, it is crucial to remember that to perform the partial fraction decomposition of a rational expression, one must often rewrite the numerator so that its degree is less than the degree of the denominator, which in this case is four because the denominator is a square. The decomposition would typically involve A/(5x^2 + 1) + Bx/(5x^2 + 1) + C/(5x^2 + 1)^2, where A, B, and C are constants or polynomials whose degrees are less than the denominators. These constants or polynomials must be determined by equating coefficients after multiplying both sides by the common denominator.
To accurately apply the partial fraction decomposition technique, a student would need to equate the numerator of the original expression with the numerators obtained after distributing the common denominator. This process involves solving for the unknown coefficients, which can be done using techniques such as substitution, equating coefficients, and sometimes the use of systems of equations.