To factorize the quadratic expression \( 9x^2 + 21x + 6 \), let's follow these steps:

1. **Identify coefficients:**
   The quadratic expression is of the form \( ax^2 + bx + c \), where:
   - \( a = 9 \)
   - \( b = 21 \)
   - \( c = 6 \)

2. **Find common factors:**
   First, notice that all terms have a common factor of 3. We can factor out this 3:
   \[
   9x^2 + 21x + 6 = 3(3x^2 + 7x + 2)
   \]

3. **Factor the quadratic inside the parentheses:**
   Now, we need to factor \( 3x^2 + 7x + 2 \). To do this, find two numbers that multiply to \( 3 \times 2 = 6 \) and add up to \( 7 \). These numbers are 6 and 1.

4. **Rewrite the middle term:**
   Rewrite \( 7x \) as \( 6x + x \):
   \[
   3x^2 + 7x + 2 = 3x^2 + 6x + x + 2
   \]

5. **Group and factor by grouping:**
   Group the terms:
   \[
   (3x^2 + 6x) + (x + 2)
   \]
   Factor each group:
   \[
   3x(x + 2) + 1(x + 2)
   \]

6. **Factor out the common binomial:**
   \[
   (3x + 1)(x + 2)
   \]

7. **Final factored form:**
   Now, include the factor of 3 we initially factored out:
   \[
   9x^2 + 21x + 6 = 3(3x + 1)(x + 2)
   \]

Thus, the fully factored form is:
\[
9x^2 + 21x + 6 = 3(3x + 1)(x + 2)
\]

Would you like more details on any of the steps?

Here are 5 related questions:
1. How do you solve quadratic equations by factoring?
2. Can you factorize \( 6x^2 + 11x + 3 \)?
3. How do you complete the square for quadratic expressions?
4. How do you use the quadratic formula to solve equations?
5. What is the difference between factoring and expanding expressions?

**Tip:** When factoring, always look for common factors first before factoring the quadratic.

Learn how to factor the quadratic expression 9x² + 21x + 6 through step-by-step guidance. This solution explains factoring by grouping, identifying common factors, and solving quadratic equations. Ideal for students in Grades 8-10.

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