Final answer:
To find the quotient using long division, divide each term of the numerator by the divisor, x + 3. The quotient is 4x² - 23x + 78 with a remainder of -234.
Step-by-step explanation:
To find the quotient using long division, we divide each term of the numerator by the divisor, x + 3. Let's divide each term one by one:
Step 1: Divide the first term, 4x³, by x. This gives us 4x².
Step 2: Multiply 4x² by the divisor, x + 3. This gives us 4x³ + 12x².
Step 3: Subtract 4x³ + 12x² from the numerator, 4x³ - 11x². This gives us -23x².
Step 4: Bring down the next term of the numerator, which is 9.
Step 5: Divide -23x² by x. This gives us -23x.
Step 6: Multiply -23x by the divisor, x + 3. This gives us -23x² - 69x.
Step 7: Subtract -23x² - 69x from the numerator, -23x² + 9. This gives us 78x.
Step 8: Bring down the next term of the numerator, which is 0.
Step 9: Divide 78x by x. This gives us 78.
Step 10: Multiply 78 by the divisor, x + 3. This gives us 78x + 234.
Step 11: Subtract 78x + 234 from the numerator, 78x. This gives us -234.
Since the remainder is -234 (a nonzero value), the division is not exact. Therefore, the quotient is 4x² - 23x + 78 with a remainder of -234. So, none of the given options (a), (b), (c), or (d) is correct.