Final answer:
To divide the polynomials P(x) = x⁵ + x⁴ - 5x³ + 4x + 4 and D(x) = x² + x - 1, we can use long division to obtain the quotient and remainder.
Step-by-step explanation:
To divide the polynomials P(x) = x⁵ + x⁴ - 5x³ + 4x + 4 and D(x) = x² + x - 1, we can use the long division method.
We start by dividing the highest degree term of P(x) by the highest degree term of D(x). In this case, it is (x⁵ / x²) = x³. This becomes the first term of the quotient.
Next, we multiply D(x) by the result from the previous step (x³) and subtract it from P(x). We repeat this process with the remainder until we obtain a degree lower than D(x) or a zero remainder.
The final quotient is x³ + 3x² + 2x + 6, and the remainder is 10x + 10.