https://file.aiquickdraw.com/mathbot/file/1725499209545dygkpznm.jpg

The expression shown in the image is:

\[
\frac{2x^2 - 32}{x^3 + 9x^2 + 20x} \cdot \frac{x^3 + 5x^2}{x^2 - 9x + 20}
\]

Let's break this down step-by-step:

### Step 1: Factor each part of the expression.
#### Factor the numerator and denominator of each fraction where possible.

1. **First numerator**: \( 2x^2 - 32 \)
   \[
   2x^2 - 32 = 2(x^2 - 16) = 2(x - 4)(x + 4)
   \]

2. **First denominator**: \( x^3 + 9x^2 + 20x \)
   Factor out \( x \):
   \[
   x(x^2 + 9x + 20) = x(x + 5)(x + 4)
   \]

3. **Second numerator**: \( x^3 + 5x^2 \)
   Factor out \( x^2 \):
   \[
   x^2(x + 5)
   \]

4. **Second denominator**: \( x^2 - 9x + 20 \)
   Factor this quadratic:
   \[
   x^2 - 9x + 20 = (x - 4)(x - 5)
   \]

### Step 2: Substitute factored forms.
Now, substitute these factored forms back into the expression:
\[
\frac{2(x - 4)(x + 4)}{x(x + 5)(x + 4)} \cdot \frac{x^2(x + 5)}{(x - 4)(x - 5)}
\]

### Step 3: Simplify the expression.
Now, cancel out common factors in the numerator and denominator:

- The \( (x - 4) \) term cancels out.
- The \( (x + 4) \) term cancels out.
- The \( (x + 5) \) term cancels out.

We are left with:
\[
\frac{2}{x(x - 5)}
\]

### Final Answer:
The simplified form of the expression is:
\[
\frac{2}{x(x - 5)}
\]

Let me know if you want more details or have any questions.

Here are 5 related questions to extend your understanding:

1. How would you approach this problem if one of the denominators had no real factors?
2. Can you simplify the expression if additional terms were added to the numerator?
3. What happens if you substitute specific values of \( x \) into the final simplified expression?
4. How would you handle this expression if it were part of a system of equations?
5. Can you identify which values of \( x \) would make this expression undefined?

**Tip**: Always check for common factors that can be canceled to simplify rational expressions efficiently.

Simplify the expression: (2x^2 - 32) / (x^3 + 9x^2 + 20x) * (x^3 + 5x^2) / (x^2 - 9x + 20)

Learn how to simplify the rational expression involving multiple polynomial terms by factoring and canceling out common factors. Suitable for high school students in Grades 9-12.

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