19. \( \frac{r^{2}+5 r+6}{2 r} \cdot \frac{r-2}{r+3} \)

Calculate or simplify the expression \( (r^{2}+5r+6)/(2r) * (r-2)/(r+3) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\frac{\left(r^{2}+5r+6\right)}{2r}\times \left(r-2\right)}{\left(r+3\right)}\) - step1: Remove the parentheses: \(\frac{\frac{r^{2}+5r+6}{2r}\times \left(r-2\right)}{r+3}\) - step2: Multiply the terms: \(\frac{\frac{\left(r^{2}+5r+6\right)\left(r-2\right)}{2r}}{r+3}\) - step3: Multiply by the reciprocal: \(\frac{\left(r^{2}+5r+6\right)\left(r-2\right)}{2r}\times \frac{1}{r+3}\) - step4: Rewrite the expression: \(\frac{\left(r+3\right)\left(r+2\right)\left(r-2\right)}{2r}\times \frac{1}{r+3}\) - step5: Reduce the fraction: \(\frac{\left(r+2\right)\left(r-2\right)}{2r}\times 1\) - step6: Multiply the terms: \(\frac{\left(r+2\right)\left(r-2\right)}{2r}\) - step7: Multiply the terms: \(\frac{r^{2}-4}{2r}\) The simplified form of the expression \( \frac{r^{2}+5r+6}{2r} \cdot \frac{r-2}{r+3} \) is \( \frac{r^{2}-4}{2r} \).