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To solve the expression \((\sqrt{6} - 5\sqrt{2})(5\sqrt{6} + 1)\), let's use the distributive property (also known as FOIL for binomials). Here’s a step-by-step solution:

1. **Apply Distribution (FOIL):**
   \[
   (\sqrt{6} - 5\sqrt{2})(5\sqrt{6} + 1) = \sqrt{6} \cdot 5\sqrt{6} + \sqrt{6} \cdot 1 - 5\sqrt{2} \cdot 5\sqrt{6} - 5\sqrt{2} \cdot 1
   \]

2. **Calculate Each Term:**

   - **First Term:** \(\sqrt{6} \cdot 5\sqrt{6} = 5 \cdot (\sqrt{6} \cdot \sqrt{6}) = 5 \cdot 6 = 30\)
   - **Second Term:** \(\sqrt{6} \cdot 1 = \sqrt{6}\)
   - **Third Term:** \(-5\sqrt{2} \cdot 5\sqrt{6} = -25 \cdot (\sqrt{2} \cdot \sqrt{6}) = -25 \sqrt{12}\)
   - **Fourth Term:** \(-5\sqrt{2} \cdot 1 = -5\sqrt{2}\)

3. **Simplify the Expression:**
   - Combine all terms: \(30 + \sqrt{6} - 25\sqrt{12} - 5\sqrt{2}\).
   - Notice that \(\sqrt{12}\) can be simplified: \(\sqrt{12} = 2\sqrt{3}\).
   - Substitute this into the expression: \(30 + \sqrt{6} - 25(2\sqrt{3}) - 5\sqrt{2}\).
   - Simplify further: \(30 + \sqrt{6} - 50\sqrt{3} - 5\sqrt{2}\).

So, the simplified answer is:
\[
30 + \sqrt{6} - 50\sqrt{3} - 5\sqrt{2}
\]

Would you like more details on any part of this solution?

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Here are some related questions for practice:

1. What is the process to simplify an expression with radicals?
2. How do you apply the distributive property to binomials with radicals?
3. What steps do you take to simplify \(\sqrt{12}\) to \(2\sqrt{3}\)?
4. How can you combine like radical terms in expressions?
5. How do you handle expressions that involve different square root terms?

**Tip:** When working with radicals, always check if you can simplify the square roots for a cleaner answer.

(√6 - 5√2)(5√6 + 1) = ? (Simplify your answer. Type an exact answer, using radicals as needed.)

Learn to simplify the expression (√6 - 5√2)(5√6 + 1) with a step-by-step solution. This problem covers algebraic concepts such as the distributive property, simplification of radicals, and combining like terms, suitable for high school students.

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