If $12$, $x$, $8$ and $14$ are in proportion, then what is the mean proportional between $(x-12)$ and $(x+4)$?

Since $12,x,8,14$ are in proportion, we have:
$$\frac{12}{x}=\frac{8}{14}$$
$$x=\frac{12\times 14}{8}=\frac{168}{8}=21$$
Now, we need to find the mean proportional between $(x-12)$ and $(x+4)$.Substitute $x=21$:
$$\text{First term}=x-12=21-12=9$$
$$\text{Second term}=x+4=21+4=25$$
The mean proportional between $9$ and $25$ is:
$$\text{Mean Proportional}=\sqrt{9\times 25}=3\times 5=15$$