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Factoring is a crucial step in simplifying rational expressions. It's like finding the building blocks of an equation. When you factor, you break down a complex expression into simpler terms or factors.
In our exercise, we start by factoring both the numerator: \(12p - 240\) and the denominator: \(5p - 100\). Factoring involves finding the numbers or expressions that, when multiplied together, give the original expression.
For example, in the numerator, we can factor out the greatest common factor (GCF) which is 12. We then get:
\[12p - 240 = 12(p - 20)\]
This step simplifies our expression and sets us up for the next stages. Similarly, in the denominator, the GCF is 5:
\[5p - 100 = 5(p - 20)\]
Factoring makes the math easier to manage and reveals the common factors we need to simplify the expression further.

The greatest common factor (GCF) is the largest factor that divides two numbers. Finding the GCF is essential for simplifying expressions because it helps in breaking down the expressions to their simplest form.
In our exercise, we use the GCF to factor the numerator and the denominator. For the numerator \(12p - 240\), identifying the GCF involves looking at both terms. The numbers 12 and 240 share a common factor of 12, so we can factor out 12 from the expression:
\[12p - 240 = 12(p - 20)\]
In the denominator \(5p - 100\), the terms 5 and 100 share a common factor of 5, which allows us to factor out 5:
\[5p - 100 = 5(p - 20)\]
Understanding and using the GCF helps simplify complex rational expressions, making them easier to work with.

Canceling common factors is like trimming the fat off a problem. After factoring, if the same factor appears in both the numerator and the denominator, we can cancel it out. This step greatly simplifies the expression.
In the simplified expression from our exercise, we get:
\[\frac{12(p - 20)}{5(p - 20)}\]
Both the numerator and the denominator have a common factor of \((p - 20)\). We can cancel this common factor:
\[\frac{12(p - 20)}{5(p - 20)} = \frac{12}{5}\]
After canceling, we are left with a much simpler rational expression: \(\frac{12}{5}\). This is the final simplified form. Canceling ensures that we don't have redundant terms, making the expression as straightforward as possible.