In a technical data-modeling project, you are tasked with combining two formulas to calculate a system's total efficiency. To do this, you must rewrite the rational expressions using the Lowest Common Denominator (LCD) of $(x-5)(x+2)(x+1)$. Match each starting expression below with the specific 'missing factor' from the LCD that must be multiplied into its numerator and denominator to achieve the common denominator.

You are assisting a senior engineer in updating a structural analysis tool. You need to combine two mathematical models represented by the rational expressions $\frac{3x}{x^2-3x-10}$ and $\frac{5}{x^2+3x+2}$. To do this, you must rewrite them using their Lowest Common Denominator (LCD). Arrange the following steps in the correct order to successfully rewrite these expressions with the LCD of $(x-5)(x+2)(x+1)$.

A junior systems engineer is updating a performance model that utilizes the rational expression $\frac{5}{x^2+3x+2}$. To integrate this model with others in the system, the engineer must rewrite it using the lowest common denominator $(x-5)(x+2)(x+1)$. Which of the following is the correct equivalent expression for this model?

A project coordinator is combining two cost-estimate formulas for a construction bid. One formula uses the rational expression $\frac{3x}{x^2-3x-10}$. To rewrite this expression using the lowest common denominator $(x-5)(x+2)(x+1)$, the coordinator determines that the numerator ${}3x$ and the denominator must be multiplied by the factor $(x+1)$ to maintain an equivalent expression.

Standardizing Mathematical Expressions in System Refactoring