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Let's solve the given equation:

The problem is:

\[
4x^3 + 4x^2 - x - 1 = (2x + 1)(2x^2 + Bx - 1)
\]

We need to find the value of \( B \).

### Step 1: Expand the right-hand side

Expand the expression \( (2x + 1)(2x^2 + Bx - 1) \).

\[
(2x + 1)(2x^2 + Bx - 1) = 2x(2x^2 + Bx - 1) + 1(2x^2 + Bx - 1)
\]

First, distribute \( 2x \):

\[
2x(2x^2) = 4x^3
\]
\[
2x(Bx) = 2Bx^2
\]
\[
2x(-1) = -2x
\]

Now distribute \( 1 \):

\[
1(2x^2) = 2x^2
\]
\[
1(Bx) = Bx
\]
\[
1(-1) = -1
\]

Now, add all the terms together:

\[
4x^3 + (2Bx^2 + 2x^2) + (-2x + Bx) + (-1)
\]

Simplify:

\[
4x^3 + (2B + 2)x^2 + (B - 2)x - 1
\]

### Step 2: Compare coefficients

Now, compare the expanded form with the left-hand side of the original equation:

\[
4x^3 + 4x^2 - x - 1 = 4x^3 + (2B + 2)x^2 + (B - 2)x - 1
\]

- For \( x^3 \), the coefficients already match: \( 4 = 4 \).
- For \( x^2 \), the coefficient on the left-hand side is 4, so:
  \[
  2B + 2 = 4
  \]
  Solve for \( B \):
  \[
  2B = 4 - 2
  \]
  \[
  2B = 2
  \]
  \[
  B = 1
  \]

Thus, the value of \( B \) is \( \boxed{1} \).

### Step 3: Verify the answer

To verify, substitute \( B = 1 \) back into the equation:

\[
4x^3 + 4x^2 - x - 1 = (2x + 1)(2x^2 + x - 1)
\]

Expanding the right-hand side:

\[
(2x + 1)(2x^2 + x - 1) = 4x^3 + 4x^2 - x - 1
\]

Both sides match, confirming that \( B = 1 \).

Would you like more details on any part of the solution, or do you have further questions?

### Related questions:
1. How do you solve quadratic equations by factoring?
2. What is the general process for expanding binomials?
3. How do you compare coefficients in polynomial equations?
4. What is the distributive property, and how does it apply to binomials?
5. Can you explain how to factor trinomials involving quadratic terms?

#### Tip:
When expanding products of polynomials, always group and simplify similar terms to ensure that coefficients are compared correctly.

Given 4x^3 + 4x^2 - x - 1 = (2x + 1)(2x^2 + Bx - 1). Find the value of B.

This page details how to solve for the coefficient B in the equation 4x^3 + 4x^2 - x - 1 = (2x + 1)(2x^2 + Bx - 1) by expanding and comparing coefficients. The solution is suitable for students in Grades 9-11 and covers key algebraic concepts.

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