To solve the equation \( A^2 + B^2 + AB = 11^2 \), let's follow these steps:

1. **Simplify the given equation:**
   \[
   A^2 + B^2 + AB = 121
   \]

2. **Consider potential factorization or other algebraic approaches.**

One possible way to interpret this equation is to see if it can be factored into some standard form, or to treat it as a quadratic in terms of \( A \) or \( B \).

3. **Trying possible integer values:**
   We could try different integer pairs \((A, B)\) to see if any satisfy the equation. Let's try a few possibilities:

   - For \( A = 0 \), \( B^2 = 121 \) which gives \( B = \pm 11 \).
   - For \( A = 11 \) or \( A = -11 \), we solve the corresponding equations.

4. **Plugging in values for \( A \) and solving for \( B \):**

   - For \( A = 0 \): 
     \[
     0^2 + B^2 + 0 = 121 \implies B^2 = 121 \implies B = \pm 11.
     \]
     So, \((A, B) = (0, 11)\) and \((0, -11)\).

   - For \( A = 11 \):
     \[
     11^2 + B^2 + 11B = 121 \implies 121 + B^2 + 11B = 121 \implies B^2 + 11B = 0 \implies B(B + 11) = 0.
     \]
     This gives \( B = 0 \) or \( B = -11 \).
     So, \((A, B) = (11, 0)\) and \((11, -11)\).

   - For \( A = -11 \): 
     \[
     (-11)^2 + B^2 - 11B = 121 \implies 121 + B^2 - 11B = 121 \implies B^2 - 11B = 0 \implies B(B - 11) = 0.
     \]
     This gives \( B = 0 \) or \( B = 11 \).
     So, \((A, B) = (-11, 0)\) and \((-11, 11)\).

**Summary of solutions:**

The integer pairs \((A, B)\) satisfying the equation are:

1. \((0, 11)\)
2. \((0, -11)\)
3. \((11, 0)\)
4. \((11, -11)\)
5. \((-11, 0)\)
6. \((-11, 11)\)

Would you like more details on these solutions, or to explore other approaches?

### Related Questions:
1. How can we derive general solutions for similar quadratic equations?
2. What is the geometric interpretation of the equation \( A^2 + B^2 + AB = 121 \)?
3. How do we factor expressions of the form \( A^2 + B^2 + AB \)?
4. What are the possible real roots for quadratic equations in two variables?
5. How do we solve similar problems using the method of completing the square?

**Tip:** When dealing with equations involving multiple variables, consider trying simple integer values first to quickly identify potential solutions.

This page provides a detailed step-by-step solution to the equation A^2 + B^2 + AB = 11^2. Explore integer solutions, algebraic techniques, and the application of quadratic equations. Suitable for students in Grades 8-10.

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