(38) \( \frac{\frac{2 r+10}{r^{2}}}{\frac{r^{2}-25}{4 r}} \)

To simplify \( \frac{\frac{2 r+10}{r^{2}}}{\frac{r^{2}-25}{4 r}} \), we first need to rewrite the expression as multiplication by the reciprocal of the denominator: \[ = \frac{2r + 10}{r^2} \cdot \frac{4r}{r^2 - 25} \] Next, factor the numerator \( 2r + 10 \) and the denominator \( r^2 - 25 \): \[ 2r + 10 = 2(r + 5) \] \[ r^2 - 25 = (r - 5)(r + 5) \] Now we can substitute these factors back into the expression: \[ = \frac{2(r + 5)}{r^2} \cdot \frac{4r}{(r - 5)(r + 5)} \] Next, we can cancel \( (r + 5) \) from the numerator and the denominator (assuming \( r \neq -5 \)): \[ = \frac{2 \cdot 4r}{r^2 (r - 5)} = \frac{8r}{r^2(r - 5)} = \frac{8}{r(r - 5)} \quad \text{(assuming } r \neq 0\text{)} \] Thus, the simplified expression is: \[ \frac{8}{r(r - 5)} \] This is the final result.