https://file.aiquickdraw.com/mathbot/file/172809997371230w0h5u3.jpg

The expression shown in the image is:

\[
\frac{4x + 24}{2x + 4} \cdot \frac{x^2 + 2x}{x^2 - 36}
\]

### Step-by-Step Simplification:

#### 1. **Factor the Numerators and Denominators:**

- **First fraction:**
  \[
  \frac{4x + 24}{2x + 4}
  \]
  Factor out the common factor in both numerator and denominator:
  \[
  4x + 24 = 4(x + 6), \quad 2x + 4 = 2(x + 2)
  \]
  So the first fraction becomes:
  \[
  \frac{4(x + 6)}{2(x + 2)} = 2 \cdot \frac{x + 6}{x + 2}
  \]

- **Second fraction:**
  \[
  \frac{x^2 + 2x}{x^2 - 36}
  \]
  Factor out the common factor in the numerator and use difference of squares for the denominator:
  \[
  x^2 + 2x = x(x + 2), \quad x^2 - 36 = (x - 6)(x + 6)
  \]
  So the second fraction becomes:
  \[
  \frac{x(x + 2)}{(x - 6)(x + 6)}
  \]

#### 2. **Multiply the Fractions:**

Now, multiplying the two fractions:
\[
2 \cdot \frac{(x + 6)}{(x + 2)} \cdot \frac{x(x + 2)}{(x - 6)(x + 6)}
\]
Cancel out the common terms:
- The term \((x + 6)\) appears in both numerator and denominator, so it cancels out.
- The term \((x + 2)\) also cancels out.

Thus, the expression simplifies to:
\[
\frac{2x}{x - 6}
\]

### Final Simplified Expression:
\[
\boxed{\frac{2x}{x - 6}}
\]

Let me know if you'd like further details on any steps, and here are some related questions to consider:

1. How do you factor the difference of squares?
2. What happens if the denominator is zero in rational expressions?
3. How do you simplify more complex rational expressions?
4. Can factoring always simplify a rational expression?
5. How do you check for extraneous solutions in rational equations?

**Tip:** Always check for factors that cancel out after factoring both numerators and denominators to simplify expressions efficiently.

Given the expression (4x + 24) / (2x + 4) * (x^2 + 2x) / (x^2 - 36), which option is the correct simplification?

Learn how to simplify the expression (4x + 24)/(2x + 4) * (x^2 + 2x)/(x^2 - 36) through factoring and simplifying rational expressions. This step-by-step solution is suitable for students in Grades 9-11 and covers key concepts such as factoring and simplifying rational fractions.

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