Final answer:
The question appears to involve polynomial division, where partial fractions are unnecessary because both the numerator and divisor are second-degree polynomials, resulting in a constant quotient and possibly a constant remainder.
Step-by-step explanation:
The student seems to be asking for the division of two polynomials, resulting in a quotient and remainder, which are then expressed in terms of partial fractions. The problem is actually one of polynomial long division, after which the remainder can sometimes be expressed as a sum of simpler fractions, i.e., partial fractions. However, the division of 15x² + 52x + 43 by 3x² + 5x - 8 does not present a simplifiable remainder that would necessitate partial fractions because the degree of the numerator is equal to the degree of the denominator. Thus, this division will result in a polynomial of degree 0 (a constant) plus a remainder that is also a constant or of lower degree than the divisor and thus does not require further decomposition.