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The distributive property is a fundamental tool in algebra that helps us expand expressions and simplify calculations. It asserts that multiplying a number by a sum is the same as multiplying each addend individually and then adding the products. For any numbers or expressions a, b, and c, the distributive property states that:

\( a(b + c) = ab + ac \).

When we apply the distributive property to expressions involving binomials, we often need to multiply more than two terms. This is typically where the FOIL method (First, Outer, Inner, Last) comes into play, especially when dealing with two binomials.

In our example, \((3 - \sqrt{2})(4 - \sqrt{2})\), we distributed each term from the first binomial with each term from the second, applying the distributive property step by step. As you can see, working with the distributive property sets the groundwork for simplifying more complex algebraic expressions.

Binomials are algebraic expressions consisting of two terms joined by either a plus or minus sign. Examples include \((x + y)\) and \((a - b)\). In algebra, understanding binomials is important because they often appear in polynomial expressions and equations.

When we multiply binomials, we can not simply add or subtract their terms directly. We need specific techniques like the FOIL method or distributive property. Multiplying binomials results in another expression that might need simplification by combining like terms.

Consider the binomials in our task, \((3 - \sqrt{2})\) and \((4 - \sqrt{2})\). By using the FOIL method, we meticulously multiply each term's counterpart and organize these clearly, leading us to an answer that accurately reflects the operation.

The FOIL method stands for First, Outer, Inner, Last. It's a helpful mnemonic for remembering how to multiply two binomials. It breaks down the multiplication into manageable steps, ensuring that we don't miss any terms. Here's how it works:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms.

In our specific case, \((3 - \sqrt{2})(4 - \sqrt{2})\), we apply this method as follows:

• First: \(3 \times 4 = 12\)
• Outer: \(3 \times (-\sqrt{2}) = -3\sqrt{2}\)
• Inner: \((-\sqrt{2}) \times 4 = -4\sqrt{2}\)
• Last: \((-\sqrt{2}) \times (-\sqrt{2}) = 2\)

After applying FOIL, we combine the results, arranging and combining like terms to streamline the expression down to:

\(14 - 7\sqrt{2}\). This shows how powerful the FOIL method can be in multiplying binomials safely and accurately.