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Let's analyze the equations one by one to determine which result in infinite solutions. For infinite solutions, both sides of the equation must simplify to the same expression (e.g., \(0 = 0\)).

### 1. \(\frac{1}{2}(10x + 4) = \frac{1}{3}(6 + 15x)\)
Simplify both sides:
\[
\frac{1}{2}(10x + 4) = 5x + 2
\]
\[
\frac{1}{3}(6 + 15x) = 2 + 5x
\]
So, the equation becomes:
\[
5x + 2 = 5x + 2
\]
This is true for all values of \(x\), meaning this equation has **infinite solutions**.

### 2. \(13 - 2(x + 4) = 8x + 1\)
Simplify both sides:
\[
13 - 2(x + 4) = 13 - 2x - 8 = 5 - 2x
\]
Thus, the equation becomes:
\[
5 - 2x = 8x + 1
\]
This is a linear equation, and solving it gives:
\[
5 - 2x = 8x + 1 \quad \Rightarrow \quad 5 - 1 = 8x + 2x \quad \Rightarrow \quad 4 = 10x \quad \Rightarrow \quad x = \frac{2}{5}
\]
This equation has **one solution**, \(x = \frac{2}{5}\).

### 3. \(x + 1 = x + 2\)
Simplify:
\[
x + 1 = x + 2 \quad \Rightarrow \quad 1 = 2
\]
This is false, so the equation has **no solution**.

### 4. \(5(x - 2) = 2(10 + x)\)
Simplify both sides:
\[
5(x - 2) = 5x - 10
\]
\[
2(10 + x) = 20 + 2x
\]
Thus, the equation becomes:
\[
5x - 10 = 20 + 2x
\]
Solving this:
\[
5x - 2x = 20 + 10 \quad \Rightarrow \quad 3x = 30 \quad \Rightarrow \quad x = 10
\]
This equation has **one solution**, \(x = 10\).

### Conclusion:
The equation with infinite solutions is:
\[
\boxed{\frac{1}{2}(10x + 4) = \frac{1}{3}(6 + 15x)}
\]

Would you like further details or have any questions?

### Related questions:
1. What does it mean for an equation to have no solutions?
2. Can an equation have more than one solution but not infinite solutions?
3. How do you graphically interpret equations with infinite solutions?
4. How can we verify the number of solutions algebraically?
5. What are the conditions for a system of equations to have infinite solutions?

### Tip:
Always check if simplifying both sides of an equation leads to the same expression to quickly identify infinite solutions.

Which of the following equations result in infinite solutions?

Learn how to determine which algebraic equations result in infinite solutions through simplification. The problem involves linear equations and the conditions that lead to infinite solutions, suitable for students in Grades 8-10.

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