Let's simplify the given expression step by step. The expression is:

$\frac{(x + 3)(x - 3)}{(x - 4)(x + 5)} \cdot \frac{(x - 4)(x - 4)}{3(x + 3)}$

Step 1: Combine the fractions using multiplication.

We multiply the numerators and denominators:

$= \frac{(x + 3)(x - 3) \cdot (x - 4)(x - 4)}{(x - 4)(x + 5) \cdot 3(x + 3)}$

Step 2: Cancel common factors.

• In the numerator and denominator, we have a common factor of $(x + 3)$ and $(x - 4)$.
• Cancel these common factors:

$= \frac{(x - 3) \cdot (x - 4)}{3(x + 5)}$

Step 3: Simplify the remaining expression.

Now, we can simplify the numerator:

$(x - 3)(x - 4) = x^2 - 4x - 3x + 12 = x^2 - 7x + 12$

So, the expression becomes:

$\frac{x^2 - 7x + 12}{3(x + 5)}$

Final Simplified Expression:

$\frac{x^2 - 7x + 12}{3(x + 5)}$

This is the simplified form of the given expression.

Would you like more details, or do you have any questions?

Here are 5 related questions to expand this concept:

1. How do you perform long division on polynomials?
2. What are the different methods of factoring polynomials?
3. How do you simplify more complex rational expressions?
4. What is the difference between factoring and expanding expressions?
5. How does factoring help in solving quadratic equations?

Tip: Always look for common factors between the numerator and denominator to simplify rational expressions quickly.