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Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It is a powerful tool that allows us to represent problems and situations in a generalized format rather than using specific numbers. In algebra, we work with variables, which can represent unknown values or changing quantities, and constants that have fixed values.

In the case of polynomial long division, algebra involves dividing one polynomial by another, much like you would do with numbers, but with the added complexity of these variables and their exponents. The main objective is to simplify the given expression to make it easier to understand or to use in further calculations. Algebraic division may seem daunting at first, but by understanding the process step by step, you can master it with practice.

Simplifying expressions in algebra means to make them as straightforward and clean as possible. This often involves combining like terms, reducing fractions, and factoring, where appropriate. A key aspect to simplifying is recognizing patterns and common terms within an expression, just as we do when identifying the greatest common factor.

Consider the expression given in the exercise: the numerator of the fraction has multiple terms that appear complex. However, by breaking down each term and dividing by the common factor in the denominator, we're simplifying the problem to a more manageable state. Simplification doesn't change the value of an expression; it makes the expression easier to work with and often reveals further relationships within the math problem.

A common factor refers to a number or algebraic term that divides into each term of an expression without leaving a remainder. Finding common factors is a vital step in simplifying algebraic expressions and in operations like polynomial long division.

In the exercise, the common factor for the numerator's terms is identified as 3ab. This discovery simplifies the division process, such that each term of the numerator can be divided by this commonality, resulting in a more straightforward expression. Recognizing common factors can often turn a seemingly complicated expression into one that is much easier to deal with mathematically. It's a fundamental skill in algebra that significantly improves problem-solving efficiency.