An industrial designer is calculating the difference in volume between two cubic engine components, represented by the expression ${}27u^3 - 125v^3$. To simplify the design specifications, the designer needs to factor this expression. Which of the following is the correct factored form?

A metal fabrication specialist is calculating the difference in volume between two custom cubic storage bins, modeled by the expression ${}27u^3 - 125v^3$. To simplify the design, the specialist uses the difference of cubes factoring pattern: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. Match each part of the formula on the left with its corresponding value from the specialist's expression on the right.

A logistics manager is analyzing the difference in volume between two cubic storage units, represented by the expression ${}27u^3 - 125v^3$. When factoring this expression using the difference of cubes pattern $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, one of the resulting factors is ${}3u + 5v$.

A structural estimator is calculating the volume capacity of a custom-built storage hopper. The volume difference is represented by the expression ${}27u^3 - 125v^3$. To finalize the fabrication specifications, the estimator must factor this expression using the difference of cubes pattern. Arrange the following steps of the factoring process in the correct order.

A packaging designer is calculating the difference in volume between two shipping containers, modeled by the expression ${}27u^3 - 125v^3$. When factoring this expression using the difference of cubes pattern, the designer identifies $a = 3u$ and $b = 5v$. The resulting factored form is written as $(3u - 5v)(9u^2 + x + 25v^2)$. What is the exact value of the missing term $x$?