`4x^(2) +kx +5` is divisible by `x + 1`. The same expression is also divisible by:

Options

A. `4x+5`
B. `4x-1`
C. `4x-5`
D. `x-5`

Answer

To solve the problem, we need to determine the value of \( k \) such that the quadratic expression \( 4x^2 + kx + 5 \) is divisible by \( x + 1 \). Then we will find the other factor of the expression.

### Step-by-step Solution:

1. **Understanding Divisibility**:
Since \( 4x^2 + kx + 5 \) is divisible by \( x + 1 \), it means that when we substitute \( x = -1 \) into the expression, the result should be zero.

2. **Substituting \( x = -1 \)**:
\[ f(-1) = 4(-1)^2 + k(-1) + 5 \]
\[ = 4(1) - k + 5 \]
\[ = 4 - k + 5 \]
\[ = 9 - k \]

3. **Setting the Expression to Zero**:
Since \( f(-1) = 0 \):
\[ 9 - k = 0 \]
\[ k = 9 \]

4. **Forming the New Quadratic Expression**:
Now that we have \( k = 9 \), we can rewrite the quadratic expression:
\[ 4x^2 + 9x + 5 \]

5. **Factoring the Quadratic Expression**:
We will factor \( 4x^2 + 9x + 5 \). We look for two numbers that multiply to \( 4 \times 5 = 20 \) and add to \( 9 \). The numbers \( 4 \) and \( 5 \) satisfy this condition.

6. **Rewriting the Middle Term**:
\[ 4x^2 + 4x + 5x + 5 \]

7. **Grouping**:
\[ = (4x^2 + 4x) + (5x + 5) \]
\[ = 4x(x + 1) + 5(x + 1) \]

8. **Factoring Out the Common Term**:
\[ = (4x + 5)(x + 1) \]

9. **Identifying the Other Factor**:
The expression \( 4x^2 + 9x + 5 \) is now factored into \( (4x + 5)(x + 1) \). Thus, the other factor besides \( x + 1 \) is \( 4x + 5 \).

### Final Answer:
The same expression \( 4x^2 + kx + 5 \) is also divisible by \( 4x + 5 \).