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The distributive property is a cornerstone of algebra that allows us to multiply a single term by each term within a parenthesis. When we encounter an expression like \((a-2)(a+2)\), we distribute the multiplication across the terms.

Let's picture the first term outside the parenthesis, \(a\), reaching in and multiplying both the \(a\) and \(2\) inside the second parenthesis. Then, we follow with the second term, \(-2\), doing the same. This process forms the basis of expanding algebraic expressions and is particularly handy when simplifying equations or solving for variables.

Understanding and applying this property is crucial as it applies not just to arithmetic numbers but also to algebraic expressions, which can include variables, constants, and exponents.

In algebra, when we have two binomials, we often simplify their product using what's called the FOIL Method, which stands for First, Outer, Inner, and Last.

For the expression \((a-2)(a+2)\), here's how we apply it:

• First: Multiply the first terms of each binomial, \(a\times a\), giving us \(a^2\).
• Outer: Multiply the outer terms, \(a\times 2\), which gives \(2a\).
• Inner: Multiply the inner terms, \(-2\times a\), which results in \(-2a\).
• Last: Multiply the last terms of each binomial, \(-2\times 2\), giving \(-4\).

By following the FOIL steps, we ensure that no terms are missed when multiplying binomials. This reliable pattern helps students to remember the process and systematically work through these types of problems.

Combining like terms means to add or subtract terms that have the same variable raised to the same power. In the context of the expression \(a^2 + 2a - 2a - 4\), we notice that \(2a\) and \(-2a\) are like terms because they each contain the variable \(a\) to the first power.

We combine them by simply adding their coefficients—or the numbers in front of the variables—together. Since \(2a\) and \(-2a\) are opposites, they cancel each other out, resulting in zero. This consolidation of terms simplifies our expression to \(a^2 - 4\), which is much more manageable and cleaner to work with.

Simplification of algebraic expressions often requires various algebraic operations. These operations include addition, subtraction, multiplication, and division, applied to both constants and variables.

In the expression \((a-2)(a+2)\) after applying the FOIL method, we conducted addition and subtraction when combining like terms to simplify down to \(a^2 - 4\). It's essential to become comfortable with these operations because they form the foundation of algebra and enable you to manipulate and solve for variables in equations. Careful and orderly application of these operations is key to solving algebra problems correctly.