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.