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.