If ` ( x + 4)^(3) + ( 2 x + 1)^(3) + (2 x + 5)^(3) = ( 3 x + 12) (2 x + 1) ( 2 x + 5)` , then what is the value of x ?

Options

A. `-3`
B. `-2`
C. 2
D. 3

Answer

To solve the equation \( (x + 4)^3 + (2x + 1)^3 + (2x + 5)^3 = (3x + 12)(2x + 1)(2x + 5) \), we can follow these steps:

### Step 1: Simplify the Right Side
We start by simplifying the right side of the equation. Notice that \( 3x + 12 \) can be factored as:
\[ 3(x + 4) \]
Thus, we can rewrite the right side:
\[ (3(x + 4))(2x + 1)(2x + 5) \]

### Step 2: Let \( a = x + 4 \), \( b = 2x + 1 \), and \( c = 2x + 5 \)
We can define:
- \( a = x + 4 \)
- \( b = 2x + 1 \)
- \( c = 2x + 5 \)

### Step 3: Use the Identity for Sums of Cubes
We know that if \( a + b + c = 0 \), then:
\[ a^3 + b^3 + c^3 = 3abc \]
We need to check if \( a + b + c = 0 \).

### Step 4: Calculate \( a + b + c \)
Now, we calculate:
\[ a + b + c = (x + 4) + (2x + 1) + (2x + 5) = x + 4 + 2x + 1 + 2x + 5 = 5x + 10 \]
Setting this equal to zero gives:
\[ 5x + 10 = 0 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ 5x = -10 \implies x = -2 \]

### Conclusion
Thus, the value of \( x \) is:
\[ \boxed{-2} \]