Question: An entomologist studies the growth of a bee colony, modeled by $ rac{t^2 + 5t + 6}{t + 2} $. Simplify the expression and find its value at $ t = 4 $.

["Simplifying the Bee Colony Growth Model: An Entomologist Studies T² + 5t + 6 Divided by t + 2", "When studying bee colony growth, entomologists often analyze mathematical models that describe population dynamics over time. One such model, used to estimate colony expansion from developmental stages, is given by the rational expression:", "$$\n\frac{t^2 + 5t + 6}{t + 2}\n$$", "Understanding how to simplify this expression helps researchers interpret growth patterns more clearly and efficiently.", "### Simplifying the Expression", "We begin by factoring the numerator $ t^2 + 5t + 6 $. This quadratic expression factors into:", "$$\nt^2 + 5t + 6 = (t + 2)(t + 3)\n$$", "Now substitute this back into the original expression:", "$$\n\frac{(t + 2)(t + 3)}{t + 2}\n$$", "For all values of $ t <br/>\ne -2 $ (since division by zero is undefined), we can cancel $ t + 2 $ from the numerator and denominator:", "$$\nt + 3\n$$", "Thus, the simplified model is $ t + 3 $, representing a linear approximation of colony growth after accounting for developmental stages.", "### Evaluating at $ t = 4 $", "To predict colony size at time $ t = 4 $, substitute into the simplified expression:", "$$\nt + 3 = 4 + 3 = 7\n$$", "So, the simplified model estimates the bee colony size at four weeks to be 7 units, reflecting a steady growth pattern derived from foundational biological data.", "### Final Answer", "At $ t = 4 $, the simplified value of the bee colony growth model is:", "$$\n\boxed{7}\n$$", "This demonstrates how mathematical simplification enhances entomological analysis, turning complex expressions into clear, actionable insights for ecological research."]