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Factoring is a method used to simplify expressions in algebra. It's like taking a complex number and breaking it into simpler parts. This process is essential for simplifying rational expressions. Let's look at how it works using our exercise example.
The numerator in the expression is \(x^3 + 7x^2 - 4x\). Notice that each term contains an \(x\). This common factor, \(x\), can be factored out, making our expression simpler:
\[ x(x^2 + 7x - 4) \].
For the denominator, \(x^3 - 16x\), the same principle applies. Every term here includes an \(x\), so we factor out \(x\):
\[ x(x^2 - 16) \].
By breaking down the expression into smaller parts, or factors, we make it easier to work with.

The difference of squares is a special factoring technique. It is used specifically when an expression is in the form \(a^2 - b^2\). This can always be factored into \((a - b)(a + b)\).
Let's see how this works for our denominator. We have \(x^2 - 16\). Notice that \(16\) is \(4^2\). So, we can rewrite the expression as:
\[ x^2 - 4^2 \].
This is a perfect fit for the difference of squares rule. Thus, it can be factored into:
\[ (x - 4)(x + 4) \].
Using this approach simplifies factoring specific types of algebraic expressions. In our exercise, after factoring out the common factor \(x\) from the denominator, we apply the difference of squares:
\[ x(x - 4)(x + 4) \].
This step is crucial in making further simplification possible.

Common factors are numbers or variables that divide exactly into each term in an expression. Identifying and canceling common factors is a key skill when simplifying rational expressions.
In our example, the rational expression is first rewritten with factored forms:
\[ \frac{x(x^2 + 7x - 4)}{x(x - 4)(x + 4)} \].
Notice that \(x\) is present in both the numerator and the denominator. This \(x\) is a common factor. We can cancel it out because any number divided by itself equals one:
\[ \frac{x^2 + 7x - 4}{(x - 4)(x + 4)} \].
This leaves us with a much simpler expression. Canceling common factors is all about simplifying your work and getting to the simplest form of the expression.