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Polynomials can often look complex and intimidating, but by factoring them, you can simplify these expressions significantly. Factoring is a process of breaking down a polynomial into a product of simpler polynomials or numbers that, when multiplied together, give the original polynomial.

To factor, one typically starts by identifying a common factor or by using techniques such as the difference of squares, grouping, or trinomial factoring, as seen in the problem. Focus on recognizing patterns is key. In the case of trinomial factoring, such as with the expression \(9b^2 + 9b - 10\), we look for numbers that multiply to a certain value and add to the middle coefficient.

Once the factors are identified, the polynomial can be expressed as a product of these factors. This not only helps in simplifying complex expressions but also aids in solving polynomial equations efficiently.

Polynomial division is similar to long division used with numbers. It is a way to simplify expressions by dividing one polynomial by another. This is useful when simplifying rational expressions, especially when the numerator is more complicated than the denominator.

In our exercise, the process involved initially simplifying the numerator through factoring before the division. We managed to bring the expression \(\frac{9b^2 + 9b - 10}{3b - 2}\) into a form where cancellation of common terms (found by factoring) was possible.

Always check if the division will result in a simplification by canceling out terms. The goal here is to ease handling by reducing polynomials to their simplest forms, achieving a straightforward expression or a smaller degree polynomial.

Rational functions are fractions involving polynomials in both the numerator and the denominator. They often involve simplifications by canceling common factors, similar to how you would with numerical fractions.

In dealing with rational functions, one begins by factoring both the numerator and the denominator, if possible. After expressing them as a product of their factors, any common factors can be canceled out. The goal is to simplify the fraction to its most basic form, as was achieved in the expression \(\frac{(3b + 5)(3b - 2)}{3b - 2}\), resulting in \(3b + 5\).

This simplification makes rational functions easier to handle and helps in better understanding their behavior, especially useful in calculus and advanced algebra topics. Instead of dealing with cumbersome expressions, you end up with a cleaner and easier-to-use polynomial.