19. \( \frac{r^{2}+5 r+6}{2 r} \cdot \frac{r-2}{r+3} \)
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To simplify the expression \( \frac{r^{2}+5 r+6}{2 r} \cdot \frac{r-2}{r+3} \), let's start by factoring the numerator of the first fraction. The quadratic \( r^{2}+5 r+6 \) can be factored as: \[ r^2 + 5r + 6 = (r + 2)(r + 3) \] So, we can rewrite the expression as: \[ \frac{(r + 2)(r + 3)}{2r} \cdot \frac{r - 2}{r + 3} \] Now, we can simplify: 1. The term \( r + 3 \) in the numerator of the first fraction and the denominator of the second fraction cancels out. 2. This leaves us with: \[ \frac{(r + 2)(r - 2)}{2r} \] Next, we can expand \( (r + 2)(r - 2) \) using the difference of squares: \[ (r + 2)(r - 2) = r^{2} - 4 \] Substituting this back in gives us: \[ \frac{r^{2} - 4}{2r} \] We can factor \( r^2 - 4 \) as follows: \[ r^{2} - 4 = (r - 2)(r + 2) \] Therefore, the final simplified expression is: \[ \frac{(r - 2)(r + 2)}{2r} \] If we need to write it in an alternative form, we get: \[ \frac{(r - 2)(r + 2)}{2r} \] And that's the fully simplified result!
