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The problem in the image involves matrix operations. The given matrices are:

\[
A = \begin{bmatrix} 1 & 6 \\ -7 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 6 \\ -7 \end{bmatrix}, \quad C = \begin{bmatrix} 8 \\ -11 \end{bmatrix}
\]

You are asked to find \( X \) where \( X = A^2 - 2BC \).

### Step 1: Calculate \( A^2 \)
First, we need to calculate \( A^2 \), which is \( A \times A \).

\[
A^2 = \begin{bmatrix} 1 & 6 \\ -7 & 5 \end{bmatrix} \times \begin{bmatrix} 1 & 6 \\ -7 & 5 \end{bmatrix}
\]

Multiply the matrices:

\[
A^2 = \begin{bmatrix}
(1 \times 1 + 6 \times -7) & (1 \times 6 + 6 \times 5) \\
(-7 \times 1 + 5 \times -7) & (-7 \times 6 + 5 \times 5)
\end{bmatrix}
\]

\[
A^2 = \begin{bmatrix}
1 - 42 & 6 + 30 \\
-7 - 35 & -42 + 25
\end{bmatrix}
\]

\[
A^2 = \begin{bmatrix}
-41 & 36 \\
-42 & -17
\end{bmatrix}
\]

### Step 2: Calculate \( 2BC \)
Next, calculate \( 2BC \):

\[
2BC = 2 \times \left(\begin{bmatrix} 6 \\ -7 \end{bmatrix} \times \begin{bmatrix} 8 & -11 \end{bmatrix}\right)
\]

The multiplication of \( B \) and \( C \) gives:

\[
BC = \begin{bmatrix} 6 \times 8 & 6 \times -11 \\ -7 \times 8 & -7 \times -11 \end{bmatrix} = \begin{bmatrix} 48 & -66 \\ -56 & 77 \end{bmatrix}
\]

Now multiply by 2:

\[
2BC = \begin{bmatrix} 96 & -132 \\ -112 & 154 \end{bmatrix}
\]

### Step 3: Calculate \( X = A^2 - 2BC \)

\[
X = \begin{bmatrix} -41 & 36 \\ -42 & -17 \end{bmatrix} - \begin{bmatrix} 96 & -132 \\ -112 & 154 \end{bmatrix}
\]

Subtract the matrices:

\[
X = \begin{bmatrix} -41 - 96 & 36 + 132 \\ -42 + 112 & -17 - 154 \end{bmatrix}
\]

\[
X = \begin{bmatrix} -137 & 168 \\ 70 & -171 \end{bmatrix}
\]

### Final Answer
\[
X = \begin{bmatrix} -137 & 168 \\ 70 & -171 \end{bmatrix}
\]

Would you like any further details or clarifications on this problem?

Here are five related questions you might be interested in:
1. How do you perform matrix multiplication for larger matrices?
2. What is the significance of the determinant in matrix operations?
3. Can matrices with different dimensions be multiplied?
4. How do you find the inverse of a matrix?
5. What are some real-world applications of matrix operations?

**Tip:** When performing matrix operations, always double-check the dimensions to ensure that the multiplication is valid.

If A = [[1, 6], [-7, 5]], B = [[6], [-7]], C = [[8], [-11]], then find X when X = A^2 - 2BC.

This problem involves solving for X in the equation X = A^2 - 2BC, using given matrices A, B, and C. The solution covers matrix multiplication and subtraction, making it suitable for advanced high school or undergraduate students.

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