The value of m , so that `4x^(2) - 6x - m` is dividsible by ` x =- 3` , is exact divisor of

Options

A. 9
B. 45
C. 20
D. 54

Answer

To find the value of \( m \) such that \( 4x^2 - 6x - m \) is divisible by \( x + 3 \) (which corresponds to \( x = -3 \)), we can use polynomial long division. The remainder must be zero for the polynomial to be divisible by \( x + 3 \).

### Step-by-Step Solution:

1. **Set up the polynomial and divisor**: We want to divide \( 4x^2 - 6x - m \) by \( x + 3 \).

2. **Perform the first division**:
- Divide the leading term of the dividend \( 4x^2 \) by the leading term of the divisor \( x \):
\[ 4x^2 \div x = 4x \]
- Multiply \( 4x \) by the entire divisor \( x + 3 \):
\[ 4x(x + 3) = 4x^2 + 12x \]
- Subtract this result from the original polynomial:
\[ (4x^2 - 6x - m) - (4x^2 + 12x) = -6x - 12x - m = -18x - m \]

3. **Perform the second division**:
- Now, divide the leading term \( -18x \) by \( x \):
\[ -18x \div x = -18 \]
- Multiply \( -18 \) by the entire divisor \( x + 3 \):
\[ -18(x + 3) = -18x - 54 \]
- Subtract this from the previous result:
\[ (-18x - m) - (-18x - 54) = -m + 54 \]

4. **Set the remainder to zero**:
- For the polynomial to be divisible by \( x + 3 \), the remainder must be zero:
\[ -m + 54 = 0 \]
- Solving for \( m \):
\[ m = 54 \]

### Conclusion:
The value of \( m \) such that \( 4x^2 - 6x - m \) is divisible by \( x + 3 \) is \( m = 54 \).