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When we talk about factoring in algebra, finding a common factor is often the first step. A common factor is a number or algebraic expression that divides each term in an expression without leaving a remainder. In the exercise given, the expression is composed of two terms:

• \(9a^2b^2(3x-2y)\)
• \(-6ab(3x-2y)\)

Here, both terms contain the term \((3x-2y)\) as a factor. That's what makes it a 'common factor'. Additionally, the terms share the factors \(a\), \(b\), and \(3\). By identifying these, you can simplify the process of factoring by grouping together terms and making the expression more manageable.

Algebra deals with symbols and the rules for manipulating these symbols. In algebra, expressions are computed using a set of values known as variables, which represent unknown numbers. Understanding algebraic operations is crucial when factoring expressions, just like in the given exercise.When you are asked to factor an algebraic expression, it means you should simplify it by finding common factors among terms. In our problem, terms like \( 9a^2b^2 \) and \( -6ab \) become easier to work with when you recognize that they share a common factor, allowing you to express the original equation in a more compact form. This is accomplished by following algebraic principles that instruct us to combine and rearrange terms to make them easier to interpret and solve.

Polynomials are expressions made up of variables and coefficients, connected by addition, subtraction, multiplication, and non-negative integer exponents. Understanding the structure of polynomials can help in identifying factors and roots, which are essential steps in the factoring process.In the exercise provided, you’re dealing with a polynomial featuring two main terms. Each term is a product of constants and variables. The expression \((3x - 2y)\) is seen in both terms, signifying it plays a role as a shared polynomial factor. Successfully factoring polynomials involves recognizing these common factors and systematically breaking down complex expressions into simpler, more manageable parts. That's exactly what happens when you factor out \(3ab(3x-2y)\), leading to a clearer expression, \(3ab(3x-2y)[3ab-2]\), revealing the underlying structure of the polynomial in question.