Final answer:
To divide (4x³ – x² – x – 2) by (4x² + 3x +2) using long division, divide the first term of the numerator by the first term of the denominator, then subtract and continue the process until there are no more terms. The quotient is (x - 1/4) and the remainder is (-3x - 2).
Step-by-step explanation:
To divide (4x³ – x² – x – 2) by (4x² + 3x +2) using long division, follow these steps:
Start by dividing the first term of the numerator (4x³) by the first term of the denominator (4x²). This gives us x.
Multiply the entire denominator (4x² + 3x + 2) by x, and subtract the result from the numerator (4x³ – x² – x – 2). This gives us (4x³ - x³ = 3x³).
Bring down the next term of the numerator (-x²) and repeat the process. Divide (-x²) by (4x²) to get -1/4.
Multiply the entire denominator (4x² + 3x + 2) by -1/4, and subtract the result from the current numerator (3x³ - x²). This gives us (3x³ - x²).
Repeat this process with the remaining terms of the numerator, until there are no more terms.
The final result is the quotient (x - 1/4) with a remainder of (-3x - 2).