c.

Let’s break it down! When you multiply by , you can use the distributive property (also known as the FOIL method for binomials).

First, multiply the first terms: . Next, multiply the outer terms: . Then, the inner terms: . Finally, multiply the last terms: .

Now, combine all these results:

So, the result of ( (2+\sqrt{2}) \times(1+\sqrt{5}) = 2 + 2\sqrt{5} + \sqrt{2} + \sqrt{10}. ]

How about some fun facts? Multiplying surds can lead to interesting outcomes, blending simple integers with the complexities of irrational numbers. The presence of both and in the expression gives it a unique flavor, often showing up in geometry related to squares and pentagons!

Additionally, mastering the multiplication of surds can set a solid foundation for exploring more advanced topics, like simplifying complex expressions or even diving into algebraic identities. This skill is essential for anyone tackling higher-level math or preparing for standardized tests!