Let's simplify the expression $(1 + \sqrt{2})(3 + \sqrt{2})$.

We will use the distributive property (also known as the FOIL method for binomials):

$(1 + \sqrt{2})(3 + \sqrt{2}) = 1 \cdot 3 + 1 \cdot \sqrt{2} + \sqrt{2} \cdot 3 + \sqrt{2} \cdot \sqrt{2}$

Now, simplify each term:

$= 3 + \sqrt{2} + 3\sqrt{2} + 2$

Combine like terms:

$= 3 + 2 + \sqrt{2} + 3\sqrt{2}$

$= 5 + 4\sqrt{2}$

So, the simplified expression is:

$5 + 4\sqrt{2}$

Would you like further clarification or any additional details? Here are some related questions:

1. How would you simplify a product involving binomials with similar square roots?
2. What is the FOIL method, and how does it apply to algebraic expressions?
3. How do you simplify expressions involving square roots?
4. Can this method be applied to polynomials of higher degree?
5. How would this process change if we had a subtraction instead of addition in the original expression?

Tip: When multiplying binomials, remember to apply the distributive property (FOIL) carefully and combine like terms!