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The given expression is:

\[
\frac{x^2 + 12x + 32}{5x + 20}
\]

### Step 1: Factor the numerator and denominator

#### Numerator: \( x^2 + 12x + 32 \)
We need to factor this quadratic expression. We look for two numbers that multiply to \( 32 \) and add up to \( 12 \). These numbers are \( 4 \) and \( 8 \), so we can factor the numerator as:

\[
x^2 + 12x + 32 = (x + 4)(x + 8)
\]

#### Denominator: \( 5x + 20 \)
This can be factored by taking out the common factor \( 5 \):

\[
5x + 20 = 5(x + 4)
\]

### Step 2: Simplify the expression
Now that both the numerator and the denominator are factored, the expression becomes:

\[
\frac{(x + 4)(x + 8)}{5(x + 4)}
\]

Since \( (x + 4) \) appears in both the numerator and denominator, we can cancel it out, leaving:

\[
\frac{x + 8}{5}
\]

### Final Answer:
The simplified expression is:

\[
\frac{x + 8}{5}
\]

Would you like a more detailed explanation of any step? 

Here are 5 related questions to expand your understanding:

1. How do you factor more complex quadratics?
2. What is the general method to factor expressions like \( ax^2 + bx + c \)?
3. How can we verify that two expressions are equivalent after simplifying?
4. What happens when a quadratic does not factor nicely?
5. How can factoring help in solving equations?

**Tip**: When simplifying rational expressions, always check for common factors between the numerator and the denominator to reduce the expression easily!

Simplify the expression \( \frac{x^2 + 12x + 32}{5x + 20} \).

Learn how to simplify the expression (x^2 + 12x + 32) / (5x + 20) by factoring both the numerator and the denominator. This step-by-step solution covers factoring quadratics, canceling common factors, and simplifying rational expressions, suitable for high school students.

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