To factor the expression ${}2p^3 + 54q^3$ completely, arrange the following steps in the correct logical order.

In technical fields such as advanced manufacturing, simplifying polynomial expressions is a common step in optimizing design formulas. Suppose a volume calculation formula involves the expression ${}2p^3 + 54q^3$. To simplify it completely, you must first factor out the greatest common factor and then recall the exact structure of the 'sum of cubes' pattern. Which of the following represents the completely factored form of this expression?

In a logistics design project, you are asked to simplify the material volume expression ${}2p^3 + 54q^3$ by factoring it completely. Match each component of the factoring process with its correct value.

To factor the expression ${}2p^3 + 54q^3$ completely, the first step is to factor out the greatest common factor of ${}2$.

Identifying Factor Components in Design

As a quality assurance technician in a manufacturing plant, you are verifying the stress-testing formulas for a new dual-component bracket. The stress load formula simplifies to the expression ${}2p^3 + 54q^3$. To input this into the testing software, you must factor it completely. After factoring out the greatest common factor, you get ${}2(p^3 + 27q^3)$. Recalling the sum of cubes formula, you structure the final factored form as ${}2(p + 3q)(p^2 - ? + 9q^2)$. What specific mathematical term replaces the question mark (?) to complete the factorization? (Write your answer using only numbers and letters without spaces, such as 7xy).

Factoring Design Formulas in Quality Assurance