To simplify \(\frac{x^2 + 3x + 2}{20} \div \frac{x + 2}{10}\), we can follow these steps:

1. Rewrite the division as multiplication by the reciprocal.
2. Simplify the resulting expression.

### Step 1: Rewrite the Division as Multiplication
\[
\frac{x^2 + 3x + 2}{20} \div \frac{x + 2}{10} = \frac{x^2 + 3x + 2}{20} \times \frac{10}{x + 2}
\]

### Step 2: Simplify the Resulting Expression

1. **Factorize the numerator \(x^2 + 3x + 2\):**

   To factorize \(x^2 + 3x + 2\), we look for two numbers that multiply to 2 (constant term) and add up to 3 (coefficient of the middle term):
   - These numbers are 1 and 2.

   So, we factorize as follows:
   \[
   x^2 + 3x + 2 = (x + 1)(x + 2)
   \]

2. **Rewrite the expression with the factored numerator:**

   \[
   \frac{(x + 1)(x + 2)}{20} \times \frac{10}{x + 2}
   \]

3. **Cancel common factors:**

   Notice that \((x + 2)\) is a common factor in the numerator and the denominator:
   \[
   \frac{(x + 1)(x + 2)}{20} \times \frac{10}{x + 2} = \frac{(x + 1) \cancel{(x + 2)}}{20} \times \frac{10}{\cancel{x + 2}}
   \]

   This simplifies to:
   \[
   \frac{x + 1}{20} \times 10
   \]

4. **Simplify the multiplication:**

   \[
   \frac{x + 1}{20} \times 10 = \frac{10(x + 1)}{20} = \frac{10x + 10}{20}
   \]

5. **Reduce the fraction:**

   \[
   \frac{10x + 10}{20} = \frac{10(x + 1)}{20} = \frac{10}{20} \times (x + 1) = \frac{1}{2} (x + 1) = \frac{x + 1}{2}
   \]

So, the simplified expression is:
\[
\frac{x + 1}{2}
\]

Would you like more details or have any questions?

Here are some questions you might want to ask:
1. How do you factorize quadratic expressions?
2. How do you simplify complex rational expressions?
3. Can you provide more examples of division of rational expressions?
4. What if the numerator or denominator cannot be factored?
5. How do you handle expressions with higher-degree polynomials?

**Tip:** Always remember to rewrite division problems as multiplication by the reciprocal. This makes it easier to simplify the expression.

Learn how to simplify the rational expression (x^2 + 3x + 2)/20 divided by (x + 2)/10. Follow a step-by-step solution to factorize, cancel common factors, and simplify the expression. Suitable for high school students studying algebra and rational expressions.

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