We are solving the expression:

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

Step 1: Apply the distributive property

Distribute $2\sqrt{5}$ to each term inside the parentheses:

$2\sqrt{5} \times 3\sqrt{5} + 2\sqrt{5} \times 2$

Step 2: Simplify each term

1. Simplify $2\sqrt{5} \times 3\sqrt{5}$:

   • Multiply the coefficients: $2 \times 3 = 6$
   • Multiply the square roots: $\sqrt{5} \times \sqrt{5} = 5$
   • Combine: $6 \times 5 = 30$

2. Simplify $2\sqrt{5} \times 2$:

   • Multiply the coefficients: $2 \times 2 = 4$
   • Retain the square root: $4\sqrt{5}$

Step 3: Combine results

$30 + 4\sqrt{5}$

Final Answer:

$\boxed{30 + 4\sqrt{5}}$

Would you like further clarification or an alternative explanation? Here are some related questions:

1. How does the distributive property work in algebra?
2. Why does $\sqrt{5} \times \sqrt{5}$ equal 5?
3. How do you simplify expressions involving square roots?
4. What are common mistakes when working with square root expressions?
5. Can we approximate the decimal value of $30 + 4\sqrt{5}$?

Tip: Always simplify square roots before multiplying to reduce complexity in calculations!