`8x^(3)-27` divided by `4x^(2)+6x+9` is

Options

A. `2x+3`
B. `-2x+3`
C. `2x-3`
D. `-(2x+3)`

Answer

To solve the problem of dividing \( 8x^3 - 27 \) by \( 4x^2 + 6x + 9 \), we can follow these steps:

### Step 1: Recognize the Forms
We can recognize that \( 8x^3 - 27 \) is a difference of cubes. It can be expressed as \( (2x)^3 - 3^3 \).

### Step 2: Apply the Difference of Cubes Formula
The difference of cubes can be factored using the formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \( a = 2x \) and \( b = 3 \). Therefore, we can write:
\[ 8x^3 - 27 = (2x - 3)((2x)^2 + (2x)(3) + 3^2) \]

### Step 3: Simplify the Factored Form
Now, we simplify the expression inside the parentheses:
\[ (2x)^2 = 4x^2, \quad (2x)(3) = 6x, \quad 3^2 = 9 \]
Thus, we have:
\[ 8x^3 - 27 = (2x - 3)(4x^2 + 6x + 9) \]

### Step 4: Divide the Expression
Now we can divide \( 8x^3 - 27 \) by \( 4x^2 + 6x + 9 \):
\[ \frac{8x^3 - 27}{4x^2 + 6x + 9} = \frac{(2x - 3)(4x^2 + 6x + 9)}{4x^2 + 6x + 9} \]

### Step 5: Cancel Common Factors
Since \( 4x^2 + 6x + 9 \) is present in both the numerator and the denominator, we can cancel it out:
\[ = 2x - 3 \]

### Final Answer
Thus, the result of dividing \( 8x^3 - 27 \) by \( 4x^2 + 6x + 9 \) is:
\[ \boxed{2x - 3} \]