Let's analyze each of the given equations to determine which one has no solution. An equation with no solution typically occurs when simplifying leads to a contradiction like $a = b$, where $a \neq b$.

1. $6(5x + 4) = 3(10x + 7)$

Expand both sides: $30x + 24 = 30x + 21$ Subtract $30x$ from both sides: $24 = 21$ This is a contradiction. Therefore, this equation has no solution.

2. $18x + 17 = 2(9x + 8) + 1$

Expand the right side: $18x + 17 = 18x + 16 + 1$ Simplify: $18x + 17 = 18x + 17$ This is always true, so this equation has infinitely many solutions.

3. $7(x + 4) = 7(x - 4) + 2x$

Expand both sides: $7x + 28 = 7x - 28 + 2x$ Simplify: $7x + 28 = 9x - 28$ Subtract $7x$ from both sides: $28 = 2x - 28$ Add $28$ to both sides: $56 = 2x$ Solve for $x$: $x = 28$ This equation has one solution, $x = 28$.

4. $4(3x - 7) = 8(2x + 3)$

Expand both sides: $12x - 28 = 16x + 24$ Subtract $12x$ from both sides: $-28 = 4x + 24$ Subtract $24$ from both sides: $-52 = 4x$ Solve for $x$: $x = -13$ This equation has one solution, $x = -13$.

Conclusion:

The equation that has no solution is: $6(5x + 4) = 3(10x + 7)$ This is the first equation, as it leads to the contradiction $24 = 21$.

Do you want more details or have any questions?

Here are five related questions to explore:

1. What does it mean for an equation to have no solution?
2. How do you identify when an equation has infinitely many solutions?
3. What types of equations typically result in a single solution?
4. How can you graphically represent an equation with no solution?
5. Why does subtracting the same term from both sides help simplify an equation?

Tip: Always double-check for contradictions or true statements when simplifying an equation. This will help identify whether the equation has no solution or infinitely many solutions.