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To solve the expression \((A + B)^2\) given the matrices \(A\) and \(B\) in the image, we can follow these steps:

### 1. Matrix Addition: \(A + B\)
Given matrices \(A\) and \(B\):
\[
A = \begin{pmatrix} 
0 & -1 & 2 \\ 
-1 & 1 & 1 \\ 
3 & 0 & 4 
\end{pmatrix},
\quad
B = \begin{pmatrix} 
4 & -1 & 2 \\ 
3 & 0 & 2 \\ 
0 & -1 & 4 
\end{pmatrix}
\]
First, we calculate \(A + B\):
\[
A + B = \begin{pmatrix} 
0+4 & -1+(-1) & 2+2 \\ 
-1+3 & 1+0 & 1+2 \\ 
3+0 & 0+(-1) & 4+4 
\end{pmatrix} 
= \begin{pmatrix} 
4 & -2 & 4 \\ 
2 & 1 & 3 \\ 
3 & -1 & 8 
\end{pmatrix}
\]

### 2. Squaring the Resulting Matrix: \((A + B)^2\)
Now, square the resulting matrix \(A + B\):
\[
(A + B)^2 = (A + B) \times (A + B)
\]
\[
= \begin{pmatrix} 
4 & -2 & 4 \\ 
2 & 1 & 3 \\ 
3 & -1 & 8 
\end{pmatrix}
\times
\begin{pmatrix} 
4 & -2 & 4 \\ 
2 & 1 & 3 \\ 
3 & -1 & 8 
\end{pmatrix}
\]
To perform matrix multiplication, each element of the resulting matrix is calculated as follows:

\[
\text{First row, first column: } (4)(4) + (-2)(2) + (4)(3) = 16 - 4 + 12 = 24
\]
\[
\text{First row, second column: } (4)(-2) + (-2)(1) + (4)(-1) = -8 - 2 - 4 = -14
\]
\[
\text{First row, third column: } (4)(4) + (-2)(3) + (4)(8) = 16 - 6 + 32 = 42
\]
\[
\text{Second row, first column: } (2)(4) + (1)(2) + (3)(3) = 8 + 2 + 9 = 19
\]
\[
\text{Second row, second column: } (2)(-2) + (1)(1) + (3)(-1) = -4 + 1 - 3 = -6
\]
\[
\text{Second row, third column: } (2)(4) + (1)(3) + (3)(8) = 8 + 3 + 24 = 35
\]
\[
\text{Third row, first column: } (3)(4) + (-1)(2) + (8)(3) = 12 - 2 + 24 = 34
\]
\[
\text{Third row, second column: } (3)(-2) + (-1)(1) + (8)(-1) = -6 - 1 - 8 = -15
\]
\[
\text{Third row, third column: } (3)(4) + (-1)(3) + (8)(8) = 12 - 3 + 64 = 73
\]

Thus, the resulting matrix is:
\[
(A + B)^2 = \begin{pmatrix} 
24 & -14 & 42 \\ 
19 & -6 & 35 \\ 
34 & -15 & 73 
\end{pmatrix}
\]

Would you like any additional details or have further questions?

### Related Questions:
1. How can we determine the determinant of matrix \((A + B)^2\)?
2. What are the eigenvalues of the matrix \((A + B)\)?
3. Can you perform the matrix multiplication for different matrices \(C\) and \(D\)?
4. How does matrix squaring change if \(A\) and \(B\) are not of the same dimensions?
5. What is the effect of transposing the matrix \(A + B\) before squaring it?

### Tip:
When working with matrix multiplication, always ensure that the dimensions match; the number of columns in the first matrix must equal the number of rows in the second matrix.

Learn how to solve the expression (A+B)^2 for matrices A and B. Follow our detailed step-by-step guide with matrix addition and multiplication, ensuring accuracy in your calculations. Suitable for advanced high school students and beyond.

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