Let's work through both parts of the problem step by step:

Part 1:

$(\sqrt{x} - 2\sqrt{5})(\sqrt{x} + 2\sqrt{5})$

This is a difference of squares, which has the general form: $(a - b)(a + b) = a^2 - b^2$ In this case:

• $a = \sqrt{x}$
• $b = 2\sqrt{5}$

So, applying the difference of squares formula: $(\sqrt{x})^2 - (2\sqrt{5})^2 = x - 4 \times 5 = x - 20$ Thus, the result of the first expression is: $x - 20$

Part 2:

$(\sqrt{x} + \sqrt{5})^2$ This is a binomial square, which follows the formula: $(a + b)^2 = a^2 + 2ab + b^2$ In this case:

• $a = \sqrt{x}$
• $b = \sqrt{5}$

Expanding it: $(\sqrt{x})^2 + 2(\sqrt{x})(\sqrt{5}) + (\sqrt{5})^2$ Simplifying each term: $x + 2\sqrt{5x} + 5$ Thus, the result of the second expression is: $x + 2\sqrt{5x} + 5$

Final Answers:

1. $(\sqrt{x} - 2\sqrt{5})(\sqrt{x} + 2\sqrt{5}) = x - 20$
2. $(\sqrt{x} + \sqrt{5})^2 = x + 2\sqrt{5x} + 5$

Would you like further clarification or details?

Relative questions:

1. How would you expand $(\sqrt{x} + 3)(\sqrt{x} - 3)$?
2. What is the general form for $(a - b)^2$?
3. How do you apply the distributive property to simplify binomials?
4. What happens when you multiply two terms involving square roots like $\sqrt{a} \cdot \sqrt{b}$?
5. How does the difference of squares formula simplify certain quadratic expressions?

Tip:

Remember, when multiplying square roots, you can combine them under a single square root: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.