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The given expression to reduce is:

\[
A = (1 + \sqrt{3} + \sqrt{6} + \sqrt{2})(1 + \sqrt{6} - \sqrt{3} - \sqrt{2})
\]

We will expand this product using the distributive property (i.e., multiplying each term in the first expression with each term in the second expression):

\[
A = (1 + \sqrt{3} + \sqrt{6} + \sqrt{2}) \times (1 + \sqrt{6} - \sqrt{3} - \sqrt{2})
\]

Let's break it down step-by-step:

### 1. Multiply \(1\) by each term in the second expression:
\[
1 \times (1 + \sqrt{6} - \sqrt{3} - \sqrt{2}) = 1 + \sqrt{6} - \sqrt{3} - \sqrt{2}
\]

### 2. Multiply \(\sqrt{3}\) by each term in the second expression:
\[
\sqrt{3} \times (1 + \sqrt{6} - \sqrt{3} - \sqrt{2}) = \sqrt{3} + \sqrt{18} - 3 - \sqrt{6}
\]
Since \(\sqrt{18} = 3\sqrt{2}\), this becomes:
\[
= \sqrt{3} + 3\sqrt{2} - 3 - \sqrt{6}
\]

### 3. Multiply \(\sqrt{6}\) by each term in the second expression:
\[
\sqrt{6} \times (1 + \sqrt{6} - \sqrt{3} - \sqrt{2}) = \sqrt{6} + 6 - \sqrt{18} - \sqrt{12}
\]
Using \(\sqrt{18} = 3\sqrt{2}\) and \(\sqrt{12} = 2\sqrt{3}\), this becomes:
\[
= \sqrt{6} + 6 - 3\sqrt{2} - 2\sqrt{3}
\]

### 4. Multiply \(\sqrt{2}\) by each term in the second expression:
\[
\sqrt{2} \times (1 + \sqrt{6} - \sqrt{3} - \sqrt{2}) = \sqrt{2} + \sqrt{12} - \sqrt{6} - 2
\]
Using \(\sqrt{12} = 2\sqrt{3}\), this becomes:
\[
= \sqrt{2} + 2\sqrt{3} - \sqrt{6} - 2
\]

### 5. Now, let's combine all terms:
We now sum up all the terms from the above steps:
\[
(1 + \sqrt{6} - \sqrt{3} - \sqrt{2}) + (\sqrt{3} + 3\sqrt{2} - 3 - \sqrt{6}) + (\sqrt{6} + 6 - 3\sqrt{2} - 2\sqrt{3}) + (\sqrt{2} + 2\sqrt{3} - \sqrt{6} - 2)
\]

### 6. Group like terms:
- Constant terms: \(1 - 3 + 6 - 2 = 2\)
- Terms with \(\sqrt{2}\): \(-\sqrt{2} + 3\sqrt{2} - 3\sqrt{2} + \sqrt{2} = 0\)
- Terms with \(\sqrt{3}\): \(\sqrt{3} - \sqrt{3} - 2\sqrt{3} + 2\sqrt{3} = 0\)
- Terms with \(\sqrt{6}\): \(\sqrt{6} - \sqrt{6} + \sqrt{6} - \sqrt{6} = 0\)

All radical terms cancel out, leaving just:
\[
A = 2
\]

Thus, the reduced value of \(A\) is \( \boxed{2} \).

---

Would you like more details on the steps or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you multiply binomials with square roots?
2. What strategies can you use to simplify square roots like \(\sqrt{18}\) or \(\sqrt{12}\)?
3. Why do some terms cancel out in this type of algebraic expansion?
4. How does the distributive property work in more complex expressions like this one?
5. How can you verify that the result is correct?

**Tip**: When expanding and simplifying expressions with square roots, watch carefully for terms that can combine or cancel out to avoid mistakes.

Reduce: A = (1 + √3 + √6 + √2)(1 + √6 - √3 - √2)

Learn how to reduce the radical expression (1 + √3 + √6 + √2)(1 + √6 - √3 - √2) step-by-step. This example covers algebraic expansion and simplification of square roots, suitable for high school students in Grades 10-12.

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