Simplify $$ \frac{r s+3 s}{4-s^{2}} \div \frac{4 r s}{2-s} $$

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1. Rewrite the division as multiplication by the reciprocal:
   $$
   \frac{rs+3s}{4-s^2} \div \frac{4rs}{2-s} = \frac{rs+3s}{4-s^2} \times \frac{2-s}{4rs}
   $$

2. Factor expressions where possible. Factor $$ s $$ from the numerator of the first fraction and factor the difference of two squares in the denominator:
   $$
   rs+3s = s(r+3) \quad \text{and} \quad 4-s^2 = (2-s)(2+s)
   $$
   Thus, the expression becomes:
   $$
   \frac{s(r+3)}{(2-s)(2+s)} \times \frac{2-s}{4rs}
   $$

3. Cancel the common factor $$ (2-s) $$ in the numerator and denominator:
   $$
   \frac{s(r+3)}{(2+s) \cdot 4rs}
   $$

4. Cancel the common factor $$ s $$ in the numerator and denominator:
   $$
   \frac{r+3}{(2+s) \cdot 4r}
   $$

5. Write the final simplified expression:
   $$
   \frac{r+3}{4r(2+s)}
   $$
The simplified expression is $$ \frac{r+3}{4r(2+s)} $$.

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