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

An administrative assistant at an energy consulting firm is merging two spreadsheet formulas used to analyze solar panel efficiency. The formulas contain the rational expressions $\frac{3x}{x^2-3x-10}$ and $\frac{5}{x^2+3x+2}$. To combine these into a unified report, the assistant must rewrite both expressions using their lowest common denominator (LCD) of $(x-5)(x+2)(x+1)$.

When the assistant rewrites the first expression, $\frac{3x}{x^2-3x-10}$, by multiplying the numerator and denominator by the missing LCD factor and distributing/simplifying the numerator, the simplified algebraic expression for the new numerator is ____.