It looks like you're trying to simplify or evaluate the expression:

$\frac{a^2 - b^2}{3a^2 - 6a + 3} \times \frac{4a + 4b}{a^2 - 1}$

Let’s break it down:

Step 1: Factor the individual parts

First Fraction:

$\frac{a^2 - b^2}{3a^2 - 6a + 3}$

• The numerator $a^2 - b^2$ is a difference of squares, so it factors as: $a^2 - b^2 = (a - b)(a + b)$
• The denominator $3a^2 - 6a + 3$ has a common factor of 3: $3a^2 - 6a + 3 = 3(a^2 - 2a + 1) = 3(a - 1)^2$

Thus, the first fraction becomes: $\frac{(a - b)(a + b)}{3(a - 1)^2}$

Second Fraction:

$\frac{4a + 4b}{a^2 - 1}$

• The numerator $4a + 4b$ has a common factor of 4: $4a + 4b = 4(a + b)$
• The denominator $a^2 - 1$ is a difference of squares: $a^2 - 1 = (a - 1)(a + 1)$

Thus, the second fraction becomes: $\frac{4(a + b)}{(a - 1)(a + 1)}$

Step 2: Multiply the fractions

Now, multiply the two simplified fractions: $\frac{(a - b)(a + b)}{3(a - 1)^2} \times \frac{4(a + b)}{(a - 1)(a + 1)}$

Multiply the numerators and denominators: $\frac{(a - b)(a + b) \times 4(a + b)}{3(a - 1)^2 \times (a - 1)(a + 1)}$

Simplify:

• The numerator: $(a + b)^2(a - b) \times 4$
• The denominator: $3(a - 1)^3(a + 1)$

Thus, the expression becomes: $\frac{4(a - b)(a + b)^2}{3(a - 1)^3(a + 1)}$

Final Simplified Expression:

$\frac{4(a - b)(a + b)^2}{3(a - 1)^3(a + 1)}$

---

Let me know if you'd like more details or if you have any questions!

Additional Related Questions:

1. How do you factor a quadratic expression like $a^2 - 2a + 1$?
2. What are other examples of simplifying expressions with a difference of squares?
3. How do you handle powers of polynomials when simplifying fractions?
4. What happens if we substitute specific values of $a$ and $b$ into the expression?
5. How would you approach solving for $a$ or $b$ in such an expression?

Tip:

When simplifying expressions with fractions and factoring, always look for common factors first—this often simplifies the problem significantly.