If `A= [{:(a-b-c, 2a,2a), (2b, b-c-a, 2b),(2c, 2c, c-a-b):}]`
`= (a +b+c) (x +a +b+c)^(2), x ne 0 " and " a +b +c ne 0`, then x is equal to

Options

A. `-(a +b+c)`
B. `-2(a+b+c)`
C. `2(a+b+c)`
D. abc

Explanation

To solve the problem, we need to find the value of \( x \) given the matrix \( A \) and the equation involving the determinant of \( A \).

Let's denote the matrix \( A \) as follows:

\[ A = \begin{pmatrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{pmatrix} \]

We are given that:

\[ \text{det}(A) = (a + b + c)(x + a + b + c)^2 \]

and we need to find \( x \) under the conditions \( x \neq 0 \) and \( a + b + c \neq 0 \).

### Step 1: Calculate the Determinant of Matrix A

To find the determinant of the matrix \( A \), we can use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

Substituting the elements of matrix \( A \):

\[ \text{det}(A) = (a - b - c) \left( (b - c - a)(c - a - b) - (2b)(2c) \right) - (2a) \left( (2b)(c - a - b) - (2b)(2c) \right) + (2a) \left( (2b)(2c) - (b - c - a)(2c) \right) \]

### Step 2: Simplifying the Determinant

Now we will simplify the expression obtained from the determinant calculation. This involves expanding the terms and combining like terms.

1. Expand the first term:
\[ (b - c - a)(c - a - b) - 4bc \]
This will yield a quadratic expression in terms of \( a, b, c \).

2. Expand the second term:
\[ 2a \left( 2bc - 2b(c - a - b) \right) \]
This will also yield a quadratic expression.

3. Expand the third term:
\[ 2a \left( 4bc - 2c(b - c - a) \right) \]
This will yield another expression.

After performing these calculations, we will combine all the terms to find the determinant \( \text{det}(A) \).

### Step 3: Set the Determinant Equal to the Given Expression

Once we have the determinant \( \text{det}(A) \), we set it equal to the expression \( (a + b + c)(x + a + b + c)^2 \).

### Step 4: Solve for \( x \)

To find \( x \), we will isolate \( x \) from the equation obtained in Step 3. This may involve factoring or simplifying the expression further.

### Final Step: Conclusion

After isolating \( x \), we will arrive at the final value of \( x \).