A manufacturing quality control specialist uses the expression ${}(x - 5)^2 + 6(x - 5) + 8$ to model the variance in part dimensions, where $x$ is the measured value. To factor this expression more easily, the specialist decides to use the substitution method. What is the correct substitution for $u$ that transforms this expression into a standard quadratic trinomial?

A software developer is writing a script to optimize packaging dimensions based on a specific algorithm. The optimization model relies on simplifying the complex polynomial expression $(x - 5)^2 + 6(x - 5) + 8$. To ensure the code runs efficiently and without error, the developer factors the expression using the substitution method. Arrange the mathematical steps the developer must follow in the correct sequence to fully factor the expression.

A structural engineer is analyzing the stress on a support beam using the expression $(x - 5)^2 + 6(x - 5) + 8$, where $x$ represents the load offset. To simplify the calculations, the engineer factors the expression using the substitution method. Match each formal step of this process with its corresponding mathematical representation.

A facilities coordinator is auditing the square footage of a storage unit modeled by the expression $(x - 5)^2 + 6(x - 5) + 8$. To simplify the calculation, the coordinator uses the substitution $u = x - 5$ to factor the expression. True or False: The final simplified factored form of this model is $(x - 3)(x - 1)$.

Finalizing the Substitution Method in a Cost Model

An operations analyst at a logistics company is using the substitution method to simplify a shipping delay model represented by the polynomial $(x - 5)^2 + 6(x - 5) + 8$. They define $u = x - 5$ to rewrite the model as $u^2 + 6u + 8$. After factoring this trinomial to $(u + 2)(u + 4)$, they replace $u$ with $(x - 5)$. Once the constants inside each set of parentheses are combined, the fully simplified factored model is (x - 3)(x - \text{\___}).