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Let's break down and simplify the given expression:

\[
\sqrt{2x^3 y} \cdot \sqrt{6x^2 y^2}
\]

### Step 1: Combine the square roots
We can use the property of square roots that states:

\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\]

Applying this to the expression:

\[
\sqrt{(2x^3 y) \cdot (6x^2 y^2)} = \sqrt{2 \cdot 6 \cdot x^3 \cdot x^2 \cdot y \cdot y^2}
\]

### Step 2: Multiply the terms inside the square root

\[
\sqrt{12x^{3+2} y^{1+2}} = \sqrt{12x^5 y^3}
\]

### Step 3: Simplify the square root
Now, we need to simplify \(\sqrt{12x^5 y^3}\). Break it down into perfect squares:

\[
\sqrt{12x^5 y^3} = \sqrt{(4 \cdot 3)(x^4 \cdot x)(y^2 \cdot y)}
\]

\[
= \sqrt{4} \cdot \sqrt{3} \cdot \sqrt{x^4} \cdot \sqrt{x} \cdot \sqrt{y^2} \cdot \sqrt{y}
\]

### Step 4: Simplify each term

\[
\sqrt{4} = 2, \quad \sqrt{3} \text{ stays as } \sqrt{3}, \quad \sqrt{x^4} = x^2, \quad \sqrt{x} \text{ stays as } \sqrt{x}, \quad \sqrt{y^2} = y, \quad \sqrt{y} \text{ stays as } \sqrt{y}
\]

Putting everything together:

\[
2x^2 y \sqrt{3xy}
\]

### Final Answer:
The simplified product is:

\[
2x^2 y \sqrt{3xy}
\]

So the correct choice is the second option: \(2x^2 y \sqrt{3xy}\).

Would you like further explanation on any of the steps?

### Here are some related questions:
1. How do you multiply square roots with variables?
2. What are the properties of exponents that apply when multiplying terms?
3. How can you break down a square root into its simplest form?
4. What are the steps for simplifying square roots with both numbers and variables?
5. How do you combine like terms inside a square root?

#### Tip:
Always look for perfect squares within square roots, as they simplify directly to whole numbers or variables raised to an integer power!

Find the simplified product: \( \sqrt{2x^3y} \cdot \sqrt{6x^2y^2} \)

Learn how to simplify radicals involving variables using properties of square roots and exponents. In this problem, we simplify \( \sqrt{2x^3y} \cdot \sqrt{6x^2y^2} \) to its simplest form, suitable for students in Grades 8-10.

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