smc

Here, we are told that the two quantities are equal. Squaring both sides, we get: $a+\frac{b}{c}=a^2\ast \frac{b}{c}$. Multiply both sides by $c$: $ac+b=b{a}^2$. Looking at all answer choices, we can see that $ac=b{a}^2-b=b(a^2-1)$. This means that $c=\frac{b(a^2-1)}{a}$, and this is option C. Therefore chose $\boxed{c=\frac{b(a^2-1)}{a}}$ as your answer.