The given expression is:

$(\sqrt{5} + 5)(\sqrt{5} + 1)$

We can solve this by expanding it using the distributive property (FOIL method):

1. First term: $\sqrt{5} \times \sqrt{5} = 5$
2. Outer term: $\sqrt{5} \times 1 = \sqrt{5}$
3. Inner term: $5 \times \sqrt{5} = 5\sqrt{5}$
4. Last term: $5 \times 1 = 5$

Now, we combine the terms:

$5 + \sqrt{5} + 5\sqrt{5} + 5 = 10 + 6\sqrt{5}$

So the simplified expression is:

$10 + 6\sqrt{5}$

This corresponds to option D.

Would you like more details on this, or do you have any questions?

Here are 5 related questions to expand on the concept:

1. How do you simplify expressions involving square roots?
2. What is the FOIL method and when is it used?
3. Can you apply the distributive property to other binomial expressions involving radicals?
4. What are common mistakes when simplifying square roots?
5. How can you factor an expression like $(\sqrt{a} + b)(\sqrt{a} + c)$?

Tip: When multiplying expressions involving square roots, always simplify the square roots first if possible before applying the distributive property.