If A and B are two square matrices such that
`B=-A^(-1)BA,` then `(A+B)^(2)` is eual to-

Options

A. 0
B. `A^(2)+B^(2)`
C. `A^(2)+2AB+B^(2)`
D. `A+B`

Explanation

To solve the problem, we need to find the expression for \((A + B)^2\) given the condition \(B = -A^{-1}BA\).

### Step-by-Step Solution:

1. **Start with the given condition:**
\[ B = -A^{-1}BA \]

2. **Multiply both sides by \(A\) from the left:**
\[ AB = -BA \]

3. **Rearranging the equation gives:**
\[ AB + BA = 0 \]
This implies that:
\[ AB = -BA \]

4. **Now, we want to find \((A + B)^2\):**
\[ (A + B)^2 = A^2 + 2AB + B^2 \]

5. **Substituting \(2AB\) using the result from step 3:**
Since \(AB + BA = 0\), we can express \(2AB\) as:
\[ 2AB = -BA - AB = 0 \]
Therefore, we have:
\[ 2AB = 0 \]

6. **Substituting back into the equation for \((A + B)^2\):**
\[ (A + B)^2 = A^2 + B^2 + 0 \]
Thus, we conclude that:
\[ (A + B)^2 = A^2 + B^2 \]

### Final Result:
\[ (A + B)^2 = A^2 + B^2 \]