Answer
$$\frac{1}{\left(c-5\right)\left(5c-3\right)}$$
Work Step by Step
Recall, one never divides by a fraction. Rather, if we want to divide by a fraction, we multiply by the reciprocal of the fraction. Recall, the reciprocal of a fraction is that fraction with the numerator and the denominator switched. Thus, we find:
$$ \frac{c^2+10c+21}{\left(c^2-2c-15\right)\left(5c^2+32c-21\right)}\\ \frac{\left(c+3\right)\left(c+7\right)}{\left(c+3\right)\left(c-5\right)\left(5c-3\right)\left(c+7\right)}\\ \frac{1}{\left(c-5\right)\left(5c-3\right)}$$