Final answer:
The correct constant h in the equation 12x + 18 = h(2x + 3) for an infinite number of solutions is 6. Solving 12 = 2h gives h = 6, which satisfies both parts of the equation.
Step-by-step explanation:
To determine the value of the constant h that will result in an infinite number of solutions for the equation 12x + 18 = h(2x + 3), we need the two expressions to be equivalent. This means every term on the left side of the equation must match the corresponding term on the right side when the right side is expanded.
If we expand h(2x + 3), we get 2hx + 3h. For the equation to have infinite solutions, the coefficients of x need to match, and so do the constant terms. So, we need 12 = 2h and 18 = 3h simultaneously.
Solving the first equation, 12 = 2h, we get h = 6. Plugging h = 6 back into 18 = 3h, we get 18 = 18, which is true. Therefore, the correct answer is h = 6, which matches option A. No other values of h will satisfy both conditions and thus cannot result in an infinite number of solutions.