As a logistics analyst, you are documenting the standard procedure for optimizing your warehouse's routing software. The current calculation relies on the formula ${}27p^2q + 90pq + 75q$. To improve the algorithm's efficiency, you must input this expression in its completely factored form. Recall the standard algebraic procedure and arrange the steps of the factoring process in the correct chronological order.

In your role as a technical analyst, you are reviewing a quality control algorithm that uses the expression ${}27p^2q + 90pq + 75q$. To simplify the system's processing, you need to identify the components of its factored form. Match each mathematical component of the factoring process with its correct description.

As a technical analyst for a logistics firm, you are verifying a routing efficiency model that uses the expression ${}27p^2q + 90pq + 75q$. To ensure the software algorithm is optimized, you must record its completely factored form as specified in the standard procedural manual. Which of the following is the correct result?

Factorization Protocol Verification

As a quality assurance engineer validating a new supply chain algorithm, you refer to the standard operating procedures for code optimization. The manual states that when simplifying the initial capacity formula ${}27p^2q + 90pq + 75q$, you must first recall the procedure to extract the greatest common factor and then apply the perfect square trinomial pattern to yield the completely factored form of ${}3q(3p - 5)^2$.

Documenting Algorithm Simplification

As an administrative coordinator managing office supply budgets, you are streamlining a purchase-tracking formula represented by the expression ${}27p^2q + 90pq + 75q$. Before entering this formula into your team's budget spreadsheet, you decide to simplify it completely by factoring. Following standard operations, the very first step requires you to extract the Greatest Common Factor (GCF) from all three terms. The GCF that you must factor out first is ____.