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Polynomial factorization involves breaking down a complex polynomial into a product of simpler polynomials. This is an essential step when simplifying rational expressions. In our given exercise, factorization starts with the numerator, which is a large polynomial. The expression given is a degree 4 polynomial: \(x^{4}+x^{3}-13x^{2}-x+12\). The goal is to rewrite this polynomial as a product of two simpler polynomials. One common method to achieve this is grouping, where the terms in the polynomial are reassigned into clusters that can be more easily factored. Using this method, the original polynomial can be factored as \((x^{2}-4)(x^{2}+3x-3)\). This expresses the numerator as a product of two quadratic polynomials. Clearing up complex machinery into more manageable parts is what polynomial factorization is all about. It makes later steps, such as simplifying the fraction, much easier to perform.

In a fraction, you have a top part and a bottom part, which are crucial to understanding and simplifying rational expressions. The top part is known as the numerator, and the bottom part is the denominator.For the given exercise, the numerator is the polynomial \(x^{4}+x^{3}-13x^{2}-x+12\) before factorization, and after factorization, it becomes \((x^{2}-4)(x^{2}+3x-3)\). This process changes the appearance of the fraction but not its value.The denominator is \(x^{2}+x-12\). Once factorized, it becomes \((x-3)(x+4)\). Understanding the structure of the numerator and denominator helps us in our next steps to look for any common elements that might be canceled out to simplify further.

After polynomials in both the numerator and denominator have been factorized, we arrive at a crucial stage in simplification: canceling common factors. This step involves looking for factors that are present in both the numerator and the denominator and canceling them out. In our expression, the factorized numerator \((x^{2}-4)(x^{2}+3x-3)\) and the denominator \((x-3)(x+4)\) do not have any common factors. This means that no further simplification can be done by canceling terms. Even though no factors were canceled in this instance, always remember that identifying and canceling common factors is what often leads to a simpler expression. It’s a step not to be overlooked, even if the result doesn't change significantly. In more complex expressions, canceling common factors can significantly reduce complexity.