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When faced with expressions or numbers, one useful technique is to find the Greatest Common Factor (GCF). The GCF is the largest factor that two or more numbers or terms have in common. For expressions, it involves identifying the common divisors of the terms within the expression.

Imagine you have two numbers, like 8 and 12. Their factors are:

• 8: 1, 2, 4, 8
• 12: 1, 2, 3, 4, 6, 12

The common factors are 1, 2, and 4. Among these, the greatest is 4, making it the GCF.

For algebraic expressions, finding the GCF is similar but involves variables and coefficients. In the exercise expression \(a(9c+4) - b(9c+4)\), \((9c+4)\) is the common factor for both terms. By factoring it out, calculations and simplifications become easier.

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. It's the language of mathematics that allows us to express general mathematical ideas on how different objects relate.

Algebraic expressions are key components, composed of numbers, variables (letters), and operations (like addition and subtraction). In the given expression \(a(9c+4) - b(9c+4)\), we deal with two terms involving multiplication and subtraction.

The ability to manipulate and simplify these expressions is crucial. For example, recognizing common factors can lead to simplified expressions, which are easier to evaluate and solve. In fact, utilizing the concept of factoring, we efficiently reformulate our expression into \((9c+4)(a - b)\) by recognizing the shared part of both terms.

Expanding expressions is essentially the opposite of factoring them. It involves taking a factored expression and distributing the factors across the terms to reach the original state. This step is key when checking your work in algebra.

In our problem, once the expression is factored as \((9c+4)(a-b)\), we expand it to verify correctness by distributing:

• \((9c+4) \times a\) becomes \(a(9c+4)\)
• \((9c+4) \times -b\) becomes \(-b(9c+4)\)

Adding these back together, we get the original expression \(a(9c+4) - b(9c+4)\). Expanding not only verifies the factorization but also enhances comprehension of the algebraic structure. By mastering expanding, you gain confidence that your factorization has been executed correctly.