$$  $$factor each of the following $$ 6x^2y+12xy+12y $$

Let's factor the given expression step-by-step:

### Given:
\[
6x^2y + 12xy + 12y
\]

### Step 1: Find the greatest common factor (GCF)
Each term contains a factor of **6y**. So, we can factor out $6y$.

\[
6y(x^2 + 2x + 2)
\]

### Step 2: Check if the quadratic can be factored further
We need to determine if the quadratic $x^2 + 2x + 2$ can be factored.  
Using the discriminant:
\[
\Delta = b^2 - 4ac = 2^2 - 4(1)(2) = 4 - 8 = -4
\]
Since the discriminant is negative, the quadratic $x^2 + 2x + 2$ **cannot** be factored further over the real numbers.

### Final Answer:
\[
6y(x^2 + 2x + 2)
\]

This is the fully factored form.

---

Do you need further clarification or details?  
Here are some related questions to consider:

1. What is the discriminant, and why does it determine factorability?
2. How do you factor expressions with multiple variables?
3. Can quadratic expressions always be factored?  
4. How do we factor trinomials with no common factor?
5. What are the steps for factoring completely over complex numbers?

**Tip:** Always start by factoring out the greatest common factor (GCF) before attempting other methods.

factor each of the following 6x^2y+12xy+12y

Learn how to factor the expression 6x^2y + 12xy + 12y by finding the greatest common factor (GCF) and analyzing the quadratic part. This problem covers important algebra concepts and is suitable for students in Grades 9-10.

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