As a Quality Assurance specialist reviewing a technical training manual for a logistics firm, you are verifying a sample calculation used to simplify a cost-efficiency ratio. The task is to simplify the following product: $\frac{2x}{x^2 - 7x + 12} \cdot \frac{x^2 - 9}{6x^2}$. Arrange the steps below in the correct chronological order as they should be presented in the manual.

An environmental engineer is modeling the flow rate of a filtration system using the product of two rational expressions: $\frac{2x}{x^2 - 7x + 12} \cdot \frac{x^2 - 9}{6x^2}$. After factoring all numerators and denominators to simplify the formula, which binomial factor is found in both the numerator and the denominator?

A logistics supervisor is verifying a formula simplification in a technical training manual: $\frac{2x}{x^2 - 7x + 12} \cdot \frac{x^2 - 9}{6x^2}$. In the manual's example, the two common factors that are divided out of both the numerator and the denominator are $(x - 3)$ and ${}2x$.

As a Quality Assurance technician for a manufacturing plant, you are auditing the technical documentation for a component's flow rate formula: $\frac{2x}{x^2 - 7x + 12} \cdot \frac{x^2 - 9}{6x^2}$. To ensure the manual is accurate, match each part of the original expression with its corresponding factored or simplified form as shown in the manual's worked example.

Technical Training Manual Verification

A corporate training document for data analysts includes a step-by-step example of simplifying a cost-modeling formula represented by the product of two rational expressions: $\frac{2x}{x^2 - 7x + 12} \cdot \frac{x^2 - 9}{6x^2}$. According to the documented steps, after factoring both the numerators and denominators completely to obtain $\frac{2x}{(x - 3)(x - 4)} \cdot \frac{(x - 3)(x + 3)}{6x^2}$, the final expression is simplified by dividing out the common binomial factor $(x - 3)$ and the common monomial factor ____.

Validating the Efficiency Algorithm