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Finding the common factor is a crucial step in the process of factoring an expression. A common factor is a term or expression that can be evenly divided from each term of a mathematical expression. When you factor an expression, look for terms or components that repeat across the entire expression.
In the given exercise, we have

• \( (z+2)^2 \) and
• \(-5(z+2) \).

Both terms share the monomial \((z+2)\).
Identifying common factors like \(z+2\) helps in restructuring the expression, making it simpler to work with. This method sets the stage for further factoring and simplifies solving equations or evaluating expressions later down the line.

A binomial expression is a polynomial with exactly two terms. In our solution process, recognizing binomial expressions helps in understanding the way parts of expressions interact with each other.
For example, in our exercise, once we identify \((z+2)\) as a common part of both terms, the expression transforms effectively into a simpler one: a binomial \((u^2 - 5u)\), where \(u = z + 2\).
This new form simplifies the factoring process and underscores how binomials can be manipulated by using factoring to reveal simpler expressions or solutions to equations involving those binomials.

Simplifying polynomials involves reducing expressions to their simplest forms by factoring and combining like terms. In our example, we start with the polynomial \((z+2)^2 - 5(z+2)\). By factoring out the common factor \((z+2)\), the polynomial simplifies to \((z+2)(z+2 - 5)\).
This step breaks down further to \((z+2)(z-3)\), creating a product of two binomials. The simplified expression is now much easier to analyze and use in solving equations or performing further algebraic manipulations.
This exercise exemplifies how polynomial simplification through factoring not only reveals basic structures within expressions but also aids in solving more complex problems by making them more manageable.