4 x + 12 = a x + 3 2 In the given equation, a is

Choice C is correct. If an equation has infinitely many solutions, then the two sides of the equation must be equivalent. Multiplying each side of the given equation by $2$ yields $8x+24=a\left(x+3\right)$. Since $8$ is a common factor of both terms on the left-hand side of this equation, the equation can be rewritten as $8\left(x+3\right)=a\left(x+3\right)$. The two sides of this equation are equivalent when $a=8$. Therefore, if the given equation has infinitely many solutions, the value of $a$ is $8$.

Alternate approach: If the given equation, $4x+12=\frac{a\left(x+3\right)}{2}$, has infinitely many solutions, then both sides of this equation are equal for any value of $x$. If $x=0$, then substituting $0$ for $x$ in the given equation yields $4\left(0\right)+12=\frac{a\left(0+3\right)}{2}$, or $12=\frac{3}{2}a$. Dividing both sides of this equation by $\frac{3}{2}$ yields $8=a$.

Choice A is incorrect. If the value of $a$ is $0$, the given equation is equivalent to $4x+12=0$, which has one solution, not infinitely many solutions.

Choice B is incorrect. If the value of $a$ is $3$, the given equation is equivalent to $4x+12=\frac{3\left(x+3\right)}{2}$, or $4x+12=\frac{3}{2}x+\frac{9}{2}$, which has one solution, not infinitely many solutions.

Choice D is incorrect. If the value of $a$ is $12$, the given equation is equivalent to $4x+12=\frac{12\left(x+3\right)}{2}$, or $4x+12=6x+18$, which has one solution, not infinitely many solutions.