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The distribution property, also known as the distributive law, is a key principle in algebra that allows us to simplify expressions by distributing, or "spreading out," a single term over terms inside parentheses. In our original exercise, the distribution property is applied to multiply the term outside the parentheses \(5a_2b\) by each term inside the parentheses \(a_2 - b_2\).
This step helps us eliminate the parentheses and set up our expression for further simplification. It follows the basic rule: for any numbers or variables \(a, b, \ ext{ and } c\), \(a(b + c) = ab + ac\). In our specific exercise, applying this rule gives:

• \(5a_2b imes a_2\)
• \(5a_2b imes (-b_2)\)

This property links multiplication with addition or subtraction, providing a foundational tool to simplify complex algebraic expressions.

After applying the distribution property, we move to the multiplication of polynomials, which involves multiplying terms containing variables and their powers. The primary goal is to multiply each part of one polynomial by every part of the second. In the expression \(5a_2b imes a_2 - 5a_2b imes b_2\), we tackle two separate products:

• First, multiply \(5a_2b imes a_2\). This involves multiplying the coefficients and adding the exponents of like bases, resulting in \(5a_2^2b\).
• Second, consider \(5a_2b imes (-b_2)\). Here, the term becomes \(-5a_2b^2\) after multiplying the coefficients and adjusting the exponents.

These steps showcase how understanding polynomial multiplication is crucial to simplifying algebraic expressions. Each term involves careful attention to numerical multiplication and correct handling of variable exponents.

Once the multiplication step is complete, the final task is simplifying the entire expression to its simplest form. Simplifying refers to combining like terms and making expressions as concise as possible. However, in our problem \(5a_2^2b - 5a_2b^2\), the terms do not share common factors or bases that allow further simplification.
Simplifying is often the end-goal in algebra since it clarifies the relationship between variables and reduces the complexity of equations. Sometimes, it involves factoring expressions, but where terms are unlike or do not share factors, as in this exercise, the expression itself is as simple as it can be.
Understanding when and how to stop simplifying is as crucial as the steps leading up to it, ensuring that students are not over-complicating or unnecessarily manipulating terms that are already at their most understandable form.