To determine which equation has infinite solutions, let's analyze each option step-by-step.

Step 1: Solve and simplify each equation

Option A: $9(2x - 3) = 12x + 4 + 6x$

Expand both sides: $18x - 27 = 12x + 6x + 4$ Simplify: $18x - 27 = 18x + 4$ Subtract $18x$ from both sides: $-27 = 4$ This is a contradiction, so no solutions exist for Option A.

Option B: $14x - 8 - 6x = 2(4x - 8) + 8$

Simplify both sides: $14x - 6x - 8 = 8x - 16 + 8$ $8x - 8 = 8x - 8$ Subtract $8x$ from both sides: $-8 = -8$ This is an identity (true for all values of $x$), so infinite solutions exist for Option B.

Option C: $7x + 12 = 3(5x - 9)$

Expand and simplify: $7x + 12 = 15x - 27$ Subtract $7x$ from both sides: $12 = 8x - 27$ Add 27 to both sides: $39 = 8x$ Divide by 8: $x = \frac{39}{8}$ This has one solution, so Option C has a unique solution.

Conclusion

The equation with infinite solutions is Option B.

Would you like further details or have any questions?
Here are 5 related questions to deepen your understanding:

1. How can you recognize an equation with infinite solutions without solving completely?
2. What is the difference between an identity and a contradiction in algebra?
3. How do you simplify equations with parentheses and distributive property?
4. How can you verify if a solution is correct?
5. What are other examples of equations with no solution?

Tip: An equation with infinite solutions simplifies to an identity like $a = a$, while no-solution equations simplify to contradictions like $a \neq b$.