`[{:(,x^(2)+x,-1),(,3,2):}]+[{:(,0,-1),(,-x+1,x):}]=[{:(,0,-2),(,5,1):}]` then x is equalto-

Options

A. `-1`
B. 2
C. `4`
D. No value of x

Answer

To solve the equation given by the matrices, we need to perform the addition of the two matrices and then set them equal to the third matrix. Let's break this down step by step.

### Step 1: Write down the matrices
We have the following matrices:
1. \( A = \begin{pmatrix} x^2 + x & -1 \\ 3 & 2 \end{pmatrix} \)
2. \( B = \begin{pmatrix} 0 & -1 \\ -x + 1 & x \end{pmatrix} \)
3. \( C = \begin{pmatrix} 0 & -2 \\ 5 & 1 \end{pmatrix} \)

### Step 2: Add matrices A and B
We will add matrices A and B element-wise:
\[ A + B = \begin{pmatrix} (x^2 + x) + 0 & -1 + (-1) \\ 3 + (-x + 1) & 2 + x \end{pmatrix} \]
This simplifies to:
\[ A + B = \begin{pmatrix} x^2 + x & -2 \\ 4 - x & 2 + x \end{pmatrix} \]

### Step 3: Set the sum equal to matrix C
Now we set the resulting matrix equal to matrix C:
\[ \begin{pmatrix} x^2 + x & -2 \\ 4 - x & 2 + x \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 5 & 1 \end{pmatrix} \]

### Step 4: Equate corresponding elements
From the equality of the matrices, we can form the following equations:
1. \( x^2 + x = 0 \)
2. \( -2 = -2 \) (This is always true)
3. \( 4 - x = 5 \)
4. \( 2 + x = 1 \)

### Step 5: Solve the equations
1. From \( x^2 + x = 0 \):
\[ x(x + 1) = 0 \implies x = 0 \text{ or } x = -1 \]

2. From \( 4 - x = 5 \):
\[ -x = 1 \implies x = -1 \]

3. From \( 2 + x = 1 \):
\[ x = 1 - 2 \implies x = -1 \]

### Conclusion
The consistent solution across all equations is \( x = -1 \). Therefore, the value of \( x \) is:
\[ \boxed{-1} \]