Final answer:
To find out if (x + 2) is a factor of the given polynomial, we use the Remainder Theorem and evaluate f(-2). The remainder is 2, implying (x + 2) is not a factor, an option not listed among the multiple choices provided. Hence, a correct option is not provided among the listed answers.
Step-by-step explanation:
To determine whether (x + 2) is a factor of f(x) = 3x⁴ + 7x³ + 3x² - x - 4, we can apply the Remainder Theorem. This theorem states that if a polynomial f(x) is divided by (x - a), the remainder is f(a). Since we want to check if (x + 2) is a factor, we can find the remainder by evaluating f(-2).
To evaluate, plug -2 into the polynomial: f(-2) = 3(-2)⁴ + 7(-2)³ + 3(-2)² - (-2) - 4. After calculating, we get: f(-2) = 3(16) - 7(8) + 3(4) + 2 - 4, which simplifies to 48 - 56 + 12 + 2 - 4. The final result is 2, which means (x + 2) is not a factor of f(x), and the remainder is 2.
The correct option is:
B. (x + 2) is not a factor, remainder is 15x³ + 27x² + 57x + 114.
C. (x + 2) is not a factor, remainder is 11x³ + 19x² + 31x + 58.
D. (x + 2) is not a factor, remainder is 7x³ + 15x² + 25x + 44.
None of these options match the remainder we found, which is simply 2.