If `4x ^(2) + x ( m +1 ) + 1` is a perfect square, then a value of m is :

Options

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

Answer

To determine the value of \( m \) such that the expression \( 4x^2 + x(m + 1) + 1 \) is a perfect square, we can follow these steps:

### Step 1: Identify the structure of a perfect square
A quadratic expression \( ax^2 + bx + c \) is a perfect square if it can be expressed in the form \( (px + q)^2 \). For our expression, we want to find conditions under which \( 4x^2 + x(m + 1) + 1 \) can be written in this form.

### Step 2: Rewrite the expression
The expression is:
\[ 4x^2 + x(m + 1) + 1 \]
We can rewrite this as:
\[ (2x)^2 + x(m + 1) + 1 \]

### Step 3: Complete the square
To complete the square, we need to express the middle term in a way that fits the perfect square format. We know that:
\[ (2x + b)^2 = 4x^2 + 4bx + b^2 \]
We want to match this with our expression. Here, \( 4b \) should equal \( m + 1 \).

### Step 4: Solve for \( b \)
From the equation \( 4b = m + 1 \), we can express \( b \) as:
\[ b = \frac{m + 1}{4} \]

### Step 5: Set up the perfect square
Now, substituting \( b \) back into the completed square form, we have:
\[ (2x + \frac{m + 1}{4})^2 - \left(\frac{m + 1}{4}\right)^2 + 1 \]
For this to be a perfect square, the term:
\[ 1 - \left(\frac{m + 1}{4}\right)^2 \]
must equal zero.

### Step 6: Solve the equation
Setting the equation to zero gives us:
\[ 1 - \left(\frac{m + 1}{4}\right)^2 = 0 \]
This simplifies to:
\[ \left(\frac{m + 1}{4}\right)^2 = 1 \]
Taking the square root of both sides, we get:
\[ \frac{m + 1}{4} = \pm 1 \]

### Step 7: Solve for \( m \)
This leads to two cases:
1. \( \frac{m + 1}{4} = 1 \)
- \( m + 1 = 4 \)
- \( m = 3 \)

2. \( \frac{m + 1}{4} = -1 \)
- \( m + 1 = -4 \)
- \( m = -5 \)

### Conclusion
Thus, the possible values of \( m \) are \( 3 \) and \( -5 \). Since the question asks for a value of \( m \), we can state that one value of \( m \) is:
\[ \boxed{3} \]