Let's simplify the expression step by step:

The expression is:

$\frac{(x + 4)(x + 3)}{x(x + 3)} \times \frac{x(x + 2)}{x + 4}$

Step 1: Factorization and Cancellation

1. Identify common factors in the numerator and denominator:

   • The first fraction has $(x + 4)$ and $(x + 3)$ in the numerator and $x(x + 3)$ in the denominator.
   • The second fraction has $x(x + 2)$ in the numerator and $(x + 4)$ in the denominator.

2. Cancel out common terms:

   • $(x + 3)$ in the numerator and denominator of the first fraction can be canceled.
   • $(x + 4)$ in the numerator and denominator of the second fraction can be canceled.

After cancellation, the expression simplifies to:

$\frac{1 \times x(x + 2)}{x}$

Step 2: Simplify the Expression

Now, simplify the remaining terms:

$\frac{x(x + 2)}{x}$

Cancel $x$ from the numerator and the denominator:

$x + 2$

Final Answer

The simplified expression is:

$x + 2$

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you simplify expressions involving complex fractions?
2. What are common techniques for factoring polynomials?
3. How do you cancel out terms in a fraction properly?
4. Can you explain how to handle rational expressions with multiple variables?
5. What happens if a factor in the denominator is equal to zero?

Tip: Always check for common factors in the numerator and denominator before performing any operations in a rational expression.