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Factoring polynomials is an essential skill in algebra. It is the process of breaking down a polynomial into products of other polynomials that are simpler and easier to work with. In this exercise, we factor the denominator \( x^2 - x - 12 \) into \((x-4)(x+3)\). Understanding how to factor this correctly makes it much easier to find common denominators and simplify fractions. To factor a polynomial like \(x^2 - x - 12\), you need to find two numbers that multiply to the constant term (-12) and add to the coefficient of the x term (-1). Here, those numbers are -4 and +3. Thus, we can express \(x^2 - x - 12\) as \((x-4)(x+3)\). Breaking down polynomials like this reveals their basic components, allowing for straightforward simplification.

Finding a common denominator is another critical step in simplifying algebraic fractions. A common denominator is a shared multiple of the denominators of two or more fractions. In our exercise, both fractions already have the same denominator: \( x^2 - x - 12 \), which was factored into \((x-4)(x+3)\). Having a common denominator allows us to combine fractions into a single fraction, which simplifies the operation significantly. Without a common denominator, adding or subtracting fractions would require additional steps to make the denominators the same. Combining fractions over a common denominator streamlines the process and makes the algebra more manageable.

Reducing fractions means simplifying them to their lowest terms. This involves canceling out common factors in the numerator and denominator. In our exercise, the fraction was simplified to \( \frac{(x-5)(x+3)}{(x-4)(x+3)} \). Since \(x+3\) is a common factor in both the numerator and denominator, they can be canceled out. This leaves us with the simplified form \( \frac{x-5}{x-4} \). Reducing fractions helps to present the result in its simplest, most understandable form, making it easier to interpret and use in further calculations. Remember to cancel common factors only if they appear as a whole term in both the numerator and denominator.

Elementary algebra is the foundation of all algebraic problems and involves basic operations: addition, subtraction, multiplication, and division of algebraic expressions. It also includes techniques like factoring, finding common denominators, and simplifying expressions as seen in this exercise. Elementary algebra helps build the core understanding needed for more complex algebraic manipulations. By learning to identify common factors, rewrite fractions, and simplify expressions, students develop a strong mathematical foundation. These skills are crucial for solving equations and understanding higher-level math concepts. The exercise showcases how elementary algebra techniques come together to simplify a seemingly complicated problem efficiently.