The problem is asking to "Find the product" of the two given rational expressions:

$\frac{y^2 - 4}{6y - 12} \cdot \frac{y + 6}{y + 2}$

Let's break it down step by step:

Step 1: Factor each expression

• The first numerator $y^2 - 4$ is a difference of squares and can be factored as: $y^2 - 4 = (y - 2)(y + 2)$

• The first denominator $6y - 12$ has a common factor of 6: $6y - 12 = 6(y - 2)$

Now the first fraction becomes: $\frac{(y - 2)(y + 2)}{6(y - 2)}$

• The second fraction $\frac{y + 6}{y + 2}$ cannot be simplified further.

Step 2: Simplify the product

Now multiply the two simplified expressions:

$\frac{(y - 2)(y + 2)}{6(y - 2)} \cdot \frac{y + 6}{y + 2}$

Before multiplying, notice that we can cancel the common factors:

• $y - 2$ appears in both the numerator and denominator, so we can cancel it.
• $y + 2$ appears in both the numerator and denominator, so we can cancel it as well.

After canceling, we are left with:

$\frac{1}{6} \cdot (y + 6)$

Step 3: Multiply the remaining terms

The final product is:

$\frac{y + 6}{6}$

Thus, the simplified product is:

$\frac{y + 6}{6}$

Would you like more details or have any questions about this solution?

Related Questions:

1. How do you factor a difference of squares in general?
2. Can you always cancel terms from the numerator and denominator of a rational expression?
3. What happens if you cannot factor the expression?
4. How do you handle products of rational expressions with more complex polynomials?
5. What is the importance of checking for restrictions on $y$ (like division by zero) in these types of problems?

Tip:

Always look for common factors between the numerator and denominator before multiplying rational expressions to simplify the problem quickly.