Question

1. \\(\frac{r^2 + 8r + 12}{r + 6} \cdot \frac{6r}{r + 3}\\)

Explanation:

Step1: Factor the quadratic

Factor \( r^2 + 8r + 12 \). We need two numbers that multiply to 12 and add to 8. Those numbers are 6 and 2. So, \( r^2 + 8r + 12=(r + 6)(r + 2) \).
Now the expression becomes \( \frac{(r + 6)(r + 2)}{r + 6}\cdot\frac{6r}{r + 3} \).

Step2: Cancel common factors

Cancel out the common factor \( r + 6 \) in the numerator and denominator of the first fraction.
We get \( \frac{(r + 2)}{1}\cdot\frac{6r}{r + 3} \).

Step3: Multiply the fractions

Multiply the numerators together and the denominators together: \( \frac{(r + 2)\cdot6r}{r + 3}=\frac{6r(r + 2)}{r + 3} \) (or expand the numerator: \( \frac{6r^2 + 12r}{r + 3} \)).

Answer:

\( \frac{6r(r + 2)}{r + 3} \) (or \( \frac{6r^2 + 12r}{r + 3} \))