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The problem is written in Vietnamese and can be translated as:

"Given that \( x \) and \( y \) are integers satisfying the equation:

\[
4x(2y + 3) - (2y + 3) = 15
\]

Find the maximum value of \( x + y \)."

### Step-by-Step Solution:

1. **Factor the equation**:
   \[
   4x(2y + 3) - (2y + 3) = 15
   \]
   Factor out \( (2y + 3) \):
   \[
   (2y + 3)(4x - 1) = 15
   \]

2. **Determine possible values**:
   The product \( (2y + 3)(4x - 1) = 15 \) can be factored into pairs of integers that multiply to 15:
   \[
   (1, 15), (-1, -15), (3, 5), (-3, -5), (15, 1), (-15, -1), (5, 3), (-5, -3)
   \]
   Each pair represents possible values of \( 2y + 3 \) and \( 4x - 1 \).

3. **Solve for \( x \) and \( y \)**:
   For each factor pair, solve the system of equations:
   - \( 2y + 3 = a \)
   - \( 4x - 1 = b \)

   where \( a \) and \( b \) are from the factor pairs above.

4. **Compute maximum \( x + y \)**:
   Let's go through each pair:

   - For \( (2y + 3, 4x - 1) = (1, 15) \):
     - \( 2y + 3 = 1 \Rightarrow y = -1 \)
     - \( 4x - 1 = 15 \Rightarrow x = 4 \)
     - \( x + y = 4 + (-1) = 3 \)

   - For \( (2y + 3, 4x - 1) = (-1, -15) \):
     - \( 2y + 3 = -1 \Rightarrow y = -2 \)
     - \( 4x - 1 = -15 \Rightarrow x = -3 \)
     - \( x + y = -3 + (-2) = -5 \)

   - For \( (2y + 3, 4x - 1) = (3, 5) \):
     - \( 2y + 3 = 3 \Rightarrow y = 0 \)
     - \( 4x - 1 = 5 \Rightarrow x = 2 \)
     - \( x + y = 2 + 0 = 2 \)

   - For \( (2y + 3, 4x - 1) = (-3, -5) \):
     - \( 2y + 3 = -3 \Rightarrow y = -3 \)
     - \( 4x - 1 = -5 \Rightarrow x = -1 \)
     - \( x + y = -1 + (-3) = -4 \)

   - For \( (2y + 3, 4x - 1) = (15, 1) \):
     - \( 2y + 3 = 15 \Rightarrow y = 6 \)
     - \( 4x - 1 = 1 \Rightarrow x = \frac{1 + 1}{4} = 0.5 \) (which is not an integer, so discard this solution).

   - For \( (2y + 3, 4x - 1) = (-15, -1) \):
     - \( 2y + 3 = -15 \Rightarrow y = -9 \)
     - \( 4x - 1 = -1 \Rightarrow x = 0 \)
     - \( x + y = 0 + (-9) = -9 \)

   - For \( (2y + 3, 4x - 1) = (5, 3) \):
     - \( 2y + 3 = 5 \Rightarrow y = 1 \)
     - \( 4x - 1 = 3 \Rightarrow x = 1 \)
     - \( x + y = 1 + 1 = 2 \)

   - For \( (2y + 3, 4x - 1) = (-5, -3) \):
     - \( 2y + 3 = -5 \Rightarrow y = -4 \)
     - \( 4x - 1 = -3 \Rightarrow x = -0.5 \) (which is not an integer, so discard this solution).

### Conclusion:
The maximum value of \( x + y \) is **3** when \( x = 4 \) and \( y = -1 \).

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Would you like more details, or do you have any questions about the solution?

Here are some related questions you might find interesting:
1. What other methods could you use to solve Diophantine equations?
2. How do you check if a given solution is valid for integer constraints?
3. What are factor pairs, and how can they simplify solving equations?
4. Could a similar problem be solved graphically or using matrices?
5. How does factoring help in solving non-linear equations?

**Tip**: When solving Diophantine equations, always verify that the solutions for both variables are integers!

Given integers x and y, solve the equation: 4x(2y + 3) - (2y + 3) = 15. Find the maximum value of x + y.

Solve for integers x and y in the equation 4x(2y + 3) - (2y + 3) = 15. Learn the steps involved in factoring and solving Diophantine equations, with a focus on finding the maximum value of x + y. Suitable for Grades 9-12.

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