Let's factor the expression step by step.

### Problem:
Factor the expression completely:
\[
9x^2 + 12xy + 4y^2 + 18x + 12y + 8
\]

### Solution:

1. **Group the terms**:
   To make it easier to factor, we will group the terms in pairs:
   \[
   (9x^2 + 12xy + 4y^2) + (18x + 12y + 8)
   \]

2. **Factor each group**:

   - For the first group \(9x^2 + 12xy + 4y^2\), observe that it is a perfect square trinomial:
     \[
     9x^2 + 12xy + 4y^2 = (3x + 2y)^2
     \]
   
   - For the second group \(18x + 12y + 8\), factor out the common factor of 2:
     \[
     18x + 12y + 8 = 2(9x + 6y + 4)
     \]
     Notice that \(9x + 6y + 4\) can be factored as well:
     \[
     9x + 6y + 4 = 3(3x + 2y) + 2
     \]
     However, it doesn’t factor further neatly.

3. **Combine the factors**:
   The expression can now be written as:
   \[
   (3x + 2y)^2 + 2(9x + 6y + 4)
   \]
   But this doesn't simplify any further by typical factoring methods.

Since this expression doesn't simplify nicely into further factors, we might have reached the fully factored form in terms of simple grouping. Thus, the partially factored form is:
\[
(3x + 2y)^2 + 2(9x + 6y + 4)
\]
But given the problem, if we're asked for complete factorization, the best simplified form would be this, or we'd check if other methods apply which don't simplify this further without specific substitutions or advanced techniques.

Would you like further clarification on any part of this process or more exploration into another approach?

### Related Questions:

1. What is a perfect square trinomial, and how is it factored?
2. How do you approach factoring an expression with six terms?
3. What happens if an expression cannot be neatly factored?
4. What methods are used for factoring complex polynomials?
5. How do substitution methods help in factoring more complicated expressions?

### Tip:
When dealing with multiple terms, try grouping terms to see if common patterns like perfect squares or common factors emerge before proceeding with more complex factoring strategies.

Learn how to factor the expression 9x² + 12xy + 4y² + 18x + 12y + 8 step-by-step with a detailed solution. This problem covers key algebraic concepts, including factoring perfect square trinomials and common factor methods. Suitable for advanced high school students.

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