https://file.aiquickdraw.com/mathbot/file/17283560377417047b207.jpg

### Let's analyze and solve the problem shown in the image:

1. **First Expression:**

   \[
   \left( \sqrt{x} - 5\sqrt{2} \right)\left( \sqrt{x} + 5\sqrt{2} \right)
   \]

   This expression is of the form \((a - b)(a + b)\), which can be simplified using the formula:

   \[
   (a - b)(a + b) = a^2 - b^2
   \]

   Here, \( a = \sqrt{x} \) and \( b = 5\sqrt{2} \). Let's apply the formula:

   \[
   a^2 - b^2 = (\sqrt{x})^2 - (5\sqrt{2})^2
   \]

   Simplifying further:

   \[
   x - 25 \times 2 = x - 50
   \]

   Thus, the simplified form is:

   \[
   \boxed{x - 50}
   \]

2. **Second Expression:**

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

   This is of the form \((a - b)^2\), which can be simplified using the formula:

   \[
   (a - b)^2 = a^2 - 2ab + b^2
   \]

   Here, \( a = \sqrt{x} \) and \( b = \sqrt{2} \). Let's expand:

   \[
   a^2 - 2ab + b^2 = (\sqrt{x})^2 - 2(\sqrt{x})(\sqrt{2}) + (\sqrt{2})^2
   \]

   Simplifying each term:

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

   Thus, the simplified form is:

   \[
   \boxed{x - 2\sqrt{2x} + 2}
   \]

### Final Answer:

- **First Expression: \(\boxed{x - 50}\)**
- **Second Expression: \(\boxed{x - 2\sqrt{2x} + 2}\)**

Would you like to go through any step in detail or ask additional questions?

### 5 Related Questions:
1. How can we expand and simplify other forms of binomials, such as \((a + b)(a + c)\)?
2. What is the difference between the identities for \((a + b)^2\) and \((a - b)^2\)?
3. How can we generalize the product of two binomials \((\sqrt{a} + b)(\sqrt{a} - b)\)?
4. Can the expression \((\sqrt{x} - 5\sqrt{2})(\sqrt{x} + 5\sqrt{2})\) have real roots?
5. What is the significance of the identity \((a + b)(a - b) = a^2 - b^2\) in algebraic simplification?

### Tip:
Always identify if the given expression is a special form, such as \((a - b)(a + b)\) or \((a + b)^2\), as using these identities simplifies calculations significantly.

Multiply and simplify (sqrt(x) - 5sqrt(2))(sqrt(x) + 5sqrt(2)) and (sqrt(x) - sqrt(2))^2.

Learn how to multiply and simplify expressions involving radicals, such as (sqrt(x) - 5sqrt(2))(sqrt(x) + 5sqrt(2)) and (sqrt(x) - sqrt(2))^2, using the difference of squares and binomial expansion formulas. Suitable for Grades 9-10.

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