A machine efficiency analyst uses the function $f(x) = x^2 + 2x - 2$ to model output capacity based on the speed $x$. To determine the specific speeds where the output is exactly 6 units, the analyst must identify the corresponding points on the graph of the function. Arrange the following steps in the correct logical order to solve this problem.

A supply chain analyst uses the cost function $f(x) = x^2 + 2x - 2$ to model daily operational costs (in thousands of dollars) based on a routing variable $x$. The analyst needs to report the specific coordinate points on the function's graph when the operational cost reaches exactly 6 thousand dollars. Which of the following pairs of points correctly identifies these instances?

A quality control specialist uses the function $f(x) = x^2 + 2x - 2$ to monitor a manufacturing process. According to this model, when the system output $f(x)$ is exactly 6, the coordinate point (-4, 6) lies on the graph of the function.

A technical analyst is using the performance model $f(x) = x^2 + 2x - 2$ to determine the settings $x$ that result in an output of exactly 6 units. To find these values, the analyst must navigate several algebraic steps. Match each description of the solution stage with its corresponding mathematical equation or result.

Operations Performance Benchmark Analysis

An inventory manager uses the function $f(x) = x^2 + 2x - 2$ to model daily warehouse backlog levels based on the number of active sorting zones $x$. To find the sorting zone settings where the backlog is exactly 6 shipments, the manager sets the function output to 6, which yields the equation ${}6 = x^2 + 2x - 2$. Solving this equation gives two solutions: a non-viable negative value $x = -4$ and a viable positive value. The positive number of active sorting zones that yields a backlog of exact