If `(x+7)^(3)+(2x+8)^(3)+(2x+3)^(3)=3(x+7) (2x+8)(2x+3)`, then what is the value of x?
(a)`-3.6`
(b)`.6`
(c)`2.4`
(d)`-2.4`

Options

A. `-3.6`
B. `.6`
C. `2.4`
D. `-2.4`

Answer

To solve the equation \((x+7)^{3} + (2x+8)^{3} + (2x+3)^{3} = 3(x+7)(2x+8)(2x+3)\), we can use the identity for the sum of cubes. The identity states that:

\[ A^3 + B^3 + C^3 - 3ABC = (A + B + C)(A^2 + B^2 + C^2 - AB - AC - BC) \]

This equation holds true when \(A + B + C = 0\).

### Step 1: Define A, B, and C
Let:
- \(A = x + 7\)
- \(B = 2x + 8\)
- \(C = 2x + 3\)

### Step 2: Calculate A + B + C
Now we will sum \(A\), \(B\), and \(C\):

\[ A + B + C = (x + 7) + (2x + 8) + (2x + 3) \]

Combine like terms:

\[ A + B + C = x + 2x + 2x + 7 + 8 + 3 = 5x + 18 \]

### Step 3: Set A + B + C to 0
For the identity to hold, we set:

\[ 5x + 18 = 0 \]

### Step 4: Solve for x
Now, solve for \(x\):

\[ 5x = -18 \]
\[ x = -\frac{18}{5} = -3.6 \]

### Conclusion
Thus, the value of \(x\) is \(-3.6\).

### Final Answer
The correct option is (a) \(-3.6\).