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The problem involves finding the area of a rectangle where the side lengths are expressed with surds (square roots). The dimensions of the rectangle are given as:

- One side is \(4 + 2\sqrt{2}\)
- The adjacent side is \(2 + 4\sqrt{2}\)

We are asked to express the area of the rectangle in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are constants, and then determine the values of \(a\) and \(b\).

### Step-by-Step Solution:

1. **Area of a Rectangle:**
   The area \(A\) of a rectangle is the product of its length and width. So, the area is:
   \[
   A = \text{length} \times \text{width}
   \]
   Substituting the given dimensions:
   \[
   A = (4 + 2\sqrt{2}) \times (2 + 4\sqrt{2})
   \]

2. **Expanding the Product:**
   We expand this expression using the distributive property (FOIL method):
   \[
   A = (4)(2) + (4)(4\sqrt{2}) + (2\sqrt{2})(2) + (2\sqrt{2})(4\sqrt{2})
   \]
   Simplifying each term:
   \[
   A = 8 + 16\sqrt{2} + 4\sqrt{2} + 8(2)
   \]
   \[
   A = 8 + 16\sqrt{2} + 4\sqrt{2} + 16
   \]

3. **Combining Like Terms:**
   Combine the constants and the terms involving \(\sqrt{2}\):
   \[
   A = (8 + 16) + (16\sqrt{2} + 4\sqrt{2})
   \]
   \[
   A = 24 + 20\sqrt{2}
   \]

4. **Final Expression:**
   The area of the rectangle is \(A = 24 + 20\sqrt{2}\). This matches the form \(a + b\sqrt{2}\), where:
   - \(a = 24\)
   - \(b = 20\)

### Conclusion:
The values of \(a\) and \(b\) are:
- \(a = 24\)
- \(b = 20\)

Let me know if you would like further clarification or if you have any other questions! 

Here are some follow-up questions you could explore:

1. How would the solution change if one of the sides had a different value, such as \(3 + \sqrt{2}\)?
2. What would the area formula look like for a rectangle with side lengths involving cubic roots instead of square roots?
3. Can you express the area of a square using a similar technique when its side is expressed in surd form?
4. What are some common applications of surds in geometry and algebra?
5. How would the expansion change if both sides of the rectangle were negative?

**Tip:** When multiplying binomials involving surds, always remember to carefully expand and combine like terms for simplification.

Understand how to calculate the area of a rectangle with surd (square root) side lengths. Step-by-step solution provided for a rectangle with sides 4 + 2√2 and 2 + 4√2. Learn the method to express the area in the form a + b√2 and determine the values of a and b. Suitable for high school students studying geometry and algebra.

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