((x + 4)(x + 3))/(x(x + 3)) * (x(x + 2))/(x + 4)

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.

Learn how to simplify the rational expression ((x + 4)(x + 3))/(x(x + 3)) * (x(x + 2))/(x + 4) step-by-step. This problem covers key algebraic concepts such as rational expressions, factorization, and cancellation. Suitable for students in Grades 9-12.

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