The expression in the image is:

$(\sqrt{2} + \sqrt{3} + 2 + \sqrt{5})(-\sqrt{2} + \sqrt{3} + 2 - \sqrt{5})(\sqrt{10} + 2\sqrt{3})$

Let's break this down step by step.

1. First Step: Simplify the product of the first two binomials:

$(\sqrt{2} + \sqrt{3} + 2 + \sqrt{5})(-\sqrt{2} + \sqrt{3} + 2 - \sqrt{5})$

We'll use the distributive property and simplify like terms.

Step 1: Expanding the First Two Terms

We will expand this using distributive property (like FOIL for more than two terms).

Let me calculate it.The product of the first two expressions simplifies to:

$-2\sqrt{10} + 4\sqrt{3}$

Step 2: Multiply the result by $\sqrt{10} + 2\sqrt{3}$.

We now have:

$(-2\sqrt{10} + 4\sqrt{3})(\sqrt{10} + 2\sqrt{3})$

Let's expand this expression next.The final result of the entire expression is:

$4$

Thus, the correct answer is E. 4.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can the distributive property be applied to simplify complex binomial products?
2. What are some strategies to simplify square root expressions?
3. How does recognizing common terms help in simplifying algebraic expressions?
4. Can you explain the FOIL method for binomials and how it extends to larger expressions?
5. How would this problem change if we used different irrational terms like $\sqrt{7}$ or $\sqrt{11}$?

Tip: Always simplify expressions involving radicals by grouping like terms before expanding further. This can save calculation time and reduce errors.