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The FOIL method is a technique used to expand the product of two binomials. It's straightforward once you remember what FOIL stands for:

• First
• Outer
• Inner
• Last

Start by multiplying the First terms in each binomial, then the Outer terms, followed by the Inner terms, and finally the Last terms. For example, in the expression \((3 - \sqrt{2x})(3 - \sqrt{2x})\), here's how it looks:
First: Multiply the first terms: \(3 \times 3 = 9\)
Outer: Multiply the outer terms: \(3 \times (-\sqrt{2x}) = -3\sqrt{2x}\)
Inner: Multiply the inner terms: \((-\sqrt{2x}) \times 3 = -3\sqrt{2x}\)
Last: Multiply the last terms: \((-\sqrt{2x}) \times (-\sqrt{2x}) = 2x\)
By using FOIL, you're systematically covering all possible combinations of multiplication for terms in two binomials. Once you expand using FOIL, you'll need to combine like terms.

The square of a binomial refers to expressions like \((a + b)^2\) or \((a - b)^2\). When you square a binomial, you multiply it by itself. The general pattern is:

• For \( (a + b)^2 \), it's \( a^2 + 2ab + b^2 \)
• For \( (a - b)^2 \), it's \( a^2 - 2ab + b^2 \)

In the example \((3 - \sqrt{2x})^2\), you create a product of the binomial with itself:
\((3 - \sqrt{2x}) \times (3 - \sqrt{2x})\).
By understanding this pattern, you can expand such expressions more quickly without individually expanding each term. Recognizing this as a special product simplifies the expansion process.

Simplifying expressions is the process of combining like terms to make an expression more manageable and readable. After expanding an expression, look for terms that can be combined. In \(9 - 3\sqrt{2x} - 3\sqrt{2x} + 2x\), the like terms are \(-3\sqrt{2x}\) and another \(-3\sqrt{2x}\).
These can be combined to \(-6\sqrt{2x}\).
Similarly, any terms with the same variable part need to be combined, like adding ordinary numbers:

• Numerical coefficients of the same variable (e.g., term with \(x\))
• Constant terms without variables (e.g., numbers)

The final result is a simpler and more concise expression: \(9 - 6\sqrt{2x} + 2x\).
By simplifying, you ensure clearer communication of an expression’s value or formula, making it easier to interpret.