To simplify the expression \((x + \sqrt{2})^2 - 8\), let's follow these steps:

### Step 1: Expand the squared term \((x + \sqrt{2})^2\)
Use the formula for squaring a binomial \((a + b)^2 = a^2 + 2ab + b^2\).

Here, \(a = x\) and \(b = \sqrt{2}\):

\[
(x + \sqrt{2})^2 = x^2 + 2(x)(\sqrt{2}) + (\sqrt{2})^2
\]

\[
= x^2 + 2x\sqrt{2} + 2
\]

### Step 2: Subtract 8 from the expanded expression
Now, substitute the expanded expression back and subtract 8:

\[
x^2 + 2x\sqrt{2} + 2 - 8
\]

\[
= x^2 + 2x\sqrt{2} - 6
\]

### Final Simplified Expression:
The simplified form of \((x + \sqrt{2})^2 - 8\) is:

\[
x^2 + 2x\sqrt{2} - 6
\]

Would you like further details or have any questions on this? Here are some related questions:

1. How do you apply the binomial expansion formula in other cases?
2. What is the geometric meaning of the square of a binomial?
3. Can you simplify higher powers of binomials like \((x + \sqrt{2})^3\)?
4. How does this result change if we modify the expression by replacing the constant?
5. What happens if the radical term changes to a different irrational number?

**Tip:** Always remember to simplify the square root terms fully after expanding a binomial to avoid calculation errors.

This solution simplifies the expression (x + √2)² - 8 using the binomial expansion formula. It provides a detailed breakdown of the steps, explaining how to expand the binomial and simplify the resulting terms. Ideal for students in Grades 9-11 learning about algebra, radicals, and binomial expansion.

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