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The expression provided is:

\[
\left(3 \sqrt{x} + \sqrt{2}\right)^2
\]

To expand this, we can apply the formula for the square of a binomial \((a + b)^2 = a^2 + 2ab + b^2\), where:
- \(a = 3\sqrt{x}\)
- \(b = \sqrt{2}\)

### Step-by-Step Solution:

1. **Square the first term:**
   \[
   (3 \sqrt{x})^2 = 9x
   \]

2. **Multiply the two terms together and double the result:**
   \[
   2 \times (3 \sqrt{x}) \times (\sqrt{2}) = 6 \sqrt{2x}
   \]

3. **Square the second term:**
   \[
   (\sqrt{2})^2 = 2
   \]

### Putting It All Together:

Combining all terms, we get:
\[
\left(3 \sqrt{x} + \sqrt{2}\right)^2 = 9x + 6\sqrt{2x} + 2
\]

Would you like further details on any part of this expansion? 

### Related Questions:

1. What is the general formula for expanding \((a + b)^2\)?
2. How do we simplify terms involving square roots?
3. What happens if \(x = 4\) in this expression?
4. How can we factor expressions involving square roots?
5. How would the expression change if we had \((3 \sqrt{x} - \sqrt{2})^2\)?

### Tip:
When expanding binomials with square roots, keep each square root in its own term until you can combine terms. This minimizes errors in simplification.

Expand the expression \(\left(3 \sqrt{x} + \sqrt{2}\right)^2\).

Learn how to expand the expression \((3 \sqrt{x} + \sqrt{2})^2\) using the binomial expansion formula. This detailed solution includes step-by-step calculations to find each term in the expanded form, covering key concepts in algebra and square roots. Suitable for students in Grades 9-10.

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