`(4x + 4)(ax − 1) − x^2 + 4`
In the expression above, a is a constant. If the expression is equivalent to bx, where b is a constant, what is the value of b ?

Options

A. `-5`
B. `-3`
C. 0
D. 12

Explanation

To solve the expression \((4x + 4)(ax - 1) - x^2 + 4\) and find the value of \(b\) such that the expression is equivalent to \(bx\), we will follow these steps:

### Step 1: Expand the expression
We start with the expression:
\[ (4x + 4)(ax - 1) - x^2 + 4 \]
First, we expand \((4x + 4)(ax - 1)\):
\[ = 4x \cdot ax + 4 \cdot ax - 4x \cdot 1 - 4 \cdot 1 \]
\[ = 4ax^2 + 4ax - 4x - 4 \]

### Step 2: Combine like terms
Now, we will combine this with the remaining terms in the expression:
\[ 4ax^2 + 4ax - 4x - 4 - x^2 + 4 \]
Notice that \(-4\) and \(+4\) cancel each other out:
\[ = 4ax^2 + 4ax - 4x - x^2 \]
Now, we can rearrange the terms:
\[ = (4a - 1)x^2 + (4a - 4)x \]

### Step 3: Set the expression equal to \(bx\)
We want the expression to be equivalent to \(bx\). This means that there should be no \(x^2\) term in the expression:
\[ (4a - 1)x^2 + (4a - 4)x = bx \]
For the \(x^2\) term to vanish, we set:
\[ 4a - 1 = 0 \]

### Step 4: Solve for \(a\)
From the equation \(4a - 1 = 0\):
\[ 4a = 1 \implies a = \frac{1}{4} \]

### Step 5: Substitute \(a\) back to find \(b\)
Now, we substitute \(a = \frac{1}{4}\) into the coefficient of \(x\):
\[ b = 4a - 4 \]
Substituting \(a\):
\[ b = 4 \left(\frac{1}{4}\right) - 4 = 1 - 4 = -3 \]

### Conclusion
Thus, the value of \(b\) is:
\[ \boxed{-3} \]