Simplify each rational expression. Assume all variable expressions represent positivereal numbers. (Hint: Use factoring and divide out any common factors as a first step.) [10(4x^2-9)^2 - 25x(4x^2-9)^3] / [15(4x^2-9)^6]

A rational expression is a fraction where both the numerator and the denominator are polynomials. Understanding how to manipulate these expressions is crucial for simplification, which often involves factoring and reducing common factors. In this context, recognizing the structure of the expression helps in identifying opportunities for simplification.

Factoring is the process of breaking down a polynomial into its constituent factors, which can be multiplied together to yield the original polynomial. This is essential in simplifying rational expressions, as it allows for the cancellation of common factors in the numerator and denominator. Mastery of factoring techniques, such as recognizing special products and using the distributive property, is vital.

Common factors are terms that appear in both the numerator and the denominator of a rational expression. Identifying and canceling these factors is a key step in simplifying the expression. In the given problem, recognizing that (4x^2 - 9) is a common factor will facilitate the simplification process, leading to a more manageable expression.