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.

Logistics Load-Weight Analysis

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 exactly 6 shipments is ____.