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The expression you are asked to factor is \(x^2 + 12x + 36\). This is a quadratic expression.

### Step-by-step factorization:

1. **Identify the structure**:
   The quadratic is in the form \( ax^2 + bx + c \), where \( a = 1 \), \( b = 12 \), and \( c = 36 \).

2. **Check for perfect square**:
   Notice that \( x^2 + 12x + 36 \) can be written as a perfect square trinomial:
   \[
   x^2 + 12x + 36 = (x + 6)^2
   \]
   This works because:
   - The middle term, \(12x\), is twice the product of \(6\) and \(x\) (i.e., \(2 \times x \times 6 = 12x\)).
   - The last term, \(36\), is the square of \(6\) (i.e., \(6^2 = 36\)).

3. **Final factorized form**:
   Therefore, the factorization of \( x^2 + 12x + 36 \) is:
   \[
   (x + 6)(x + 6) \quad \text{or} \quad (x + 6)^2.
   \]

So, the factorized form is \( (x + 6)^2 \).

Would you like more details on the steps or an explanation of any part of this factorization? Here are some related questions to further your understanding:

1. What is a perfect square trinomial, and how can you identify one?
2. How can you expand \( (x + 6)^2 \) to verify the factorization?
3. What are other methods to factor quadratic expressions when they are not perfect squares?
4. How can you use the quadratic formula to solve \(x^2 + 12x + 36 = 0\)?
5. What is the geometric interpretation of \( (x + 6)^2 \)?

**Tip**: Always check if a quadratic can be written as a perfect square trinomial before applying more complex methods of factorization.

Learn how to factorize the quadratic expression x^2 + 12x + 36 using the perfect square trinomial formula. This step-by-step guide breaks down the algebraic process, making it easy for students in Grades 8-10 to understand.

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