As a manufacturing technician, you are reviewing a colleague's work on optimizing an assembly line process. The optimization model requires solving the equation $(2m + 1)(m + 3) = 12m$, where $m$ represents the motor calibration setting. Arrange the required algebraic steps in the correct order to solve this equation and verify the correct settings.

A manufacturing supervisor uses the quadratic equation $(2m + 1)(m + 3) = 12m$ to model the productivity rate $m$ of a new assembly line. To solve for $m$ by factoring, the supervisor must first rewrite the equation in the standard form $am^2 + bm + c = 0$. Which of the following is the correct standard form of this productivity model?

An operations analyst at a logistics firm is solving the equation $(2m + 1)(m + 3) = 12m$ to determine the optimal number of delivery vans ($m$) to deploy during peak hours. Match each mathematical step performed by the analyst with the correct algebraic description for that process.

An industrial designer is using the equation $(2m + 1)(m + 3) = 12m$ to determine the optimal spacing for safety sensors. To solve for $m$ by factoring, the equation must be converted to standard form ($ax^2 + bx + c = 0$). After expanding the left side and subtracting ${}12m$ from both sides, the resulting equation is ${}2m^2 - 5m + 3 =$ ____.

An operations manager is solving the equation $(2m + 1)(m + 3) = 12m$ to determine the optimal multiplier ($m$) for batch processing times. The manager calculates that the correct mathematical solutions for $m$ are $m = 1$ and $m = 3$.

Documenting the Calibration Procedure

Explaining the Algebraic Process for Calibration Equations