To simplify \((5 + 6\sqrt{5})^2\), we can expand it using the formula for squaring a binomial, \((a + b)^2 = a^2 + 2ab + b^2\).

Given:
\[
a = 5 \quad \text{and} \quad b = 6\sqrt{5}
\]

1. Square \(a\): 
   \[
   a^2 = 5^2 = 25
   \]

2. Multiply \(2ab\): 
   \[
   2ab = 2 \cdot 5 \cdot 6\sqrt{5} = 60\sqrt{5}
   \]

3. Square \(b\): 
   \[
   b^2 = (6\sqrt{5})^2 = 36 \cdot 5 = 180
   \]

Now, combine these results:
\[
(5 + 6\sqrt{5})^2 = 25 + 60\sqrt{5} + 180
\]

Simplify further:
\[
= 205 + 60\sqrt{5}
\]

So, the simplified form is:
\[
\boxed{205 + 60\sqrt{5}}
\]

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

---

1. How can the distributive property be used in binomial expansion?
2. What other methods exist to simplify square roots in binomial expressions?
3. How would the result change if the middle term had subtraction instead of addition?
4. How does the formula \((a+b)^2 = a^2 + 2ab + b^2\) derive from basic multiplication principles?
5. Can you expand other similar expressions, like \((2 + \sqrt{3})^2\)?

**Tip:** Memorizing binomial expansion formulas is helpful for quickly simplifying squared terms.

This problem simplifies the expression (5 + 6√5)^2 using binomial expansion. Follow the solution to understand the step-by-step application of the formula (a + b)^2 = a^2 + 2ab + b^2, leading to the result 205 + 60√5. Suitable for Grades 10-12.

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