Final answer:
The correct polynomial division result is x² + 15x + 69 with a remainder of -274, which corresponds to option b.
Step-by-step explanation:
The student is asking for the result of the polynomial division of (x³+19x²+114x+218)/(x+4). To perform this division, we carry out long division of polynomials much like traditional long division with numbers.
We divide the first term of the numerator by the first term of the divisor: x³/x equals x². We multiply div by x² to get x³+4x² and subtract that from the numerator. This process continues until we reach the remainder term.
The correct result is x² + 15x + 69 with a remainder of -274, which gives us option b) x² + 15x + 69 - 274/(x+4).