Yes the polynomial is a multiple of 2x+1
(2x+1)(2x2+x-1)=4x3+4x2-x-1.

Polynomial Division:

To determine whether the polynomial p(x) = 4x³ + 4x² - x - 1 is a multiple of 2x + 1, we can use the polynomial division method. By dividing p(x) by 2x + 1, we can observe the remainder of the division. If the remainder is zero, then p(x) is a multiple of 2x + 1.

Polynomial Division Process:

To perform the polynomial division, we follow these steps:

Step 1: Arrange the polynomial p(x) and the divisor 2x + 1 in descending order of powers of x.

Step 2: Divide the first term of p(x) by the first term of the divisor. In this case, divide 4x³ by 2x.

Step 3: Multiply the divisor by the quotient obtained in the previous step.

Step 4: Subtract the result obtained in the previous step from p(x).

Step 5: Repeat steps 2-4 until all terms in p(x) are exhausted or the degree of the remainder is less than the degree of the divisor.

Polynomial Division of p(x) by 2x + 1:

The polynomial division of p(x) = 4x³ + 4x² - x - 1 by 2x + 1 can be represented as follows:

2x + 1 | 4x³ + 4x² - x - 1

Step 1: Arrange the terms in descending order of powers of x.

4x³ + 4x² - x - 1

Step 2: Divide the first term of p(x) by the first term of the divisor.

4x³ ÷ 2x = 2x²

Step 3: Multiply the divisor by the quotient obtained in the previous step.

(2x²)(2x + 1) = 4x³ + 2x²

Step 4: Subtract the result obtained in the previous step from p(x).

(4x³ + 4x² - x - 1) - (4x³ + 2x²) = 2x² - x - 1

Step 5: Repeat steps 2-4 until all terms in p(x) are exhausted or the degree of the remainder is less than the degree of the divisor.

By continuing the polynomial division process, we obtain:

2x + 1 | 4x³ + 4x² - x - 1
2x² - x - 1

2x + 1 | 4x³ + 4x² - x - 1
2x² - x - 1
--------------
0

Conclusion:

Since the remainder of the polynomial division is zero, we can conclude that p(x) = 4x³ + 4x² - x - 1 is a multiple of 2x + 1.