ThenQuestion: A plant biologist is studying a population of genetically modified wheat plants. The number of plants that survive after $ t $ weeks is modeled by the function $ S(t) = rac{t^2 + 5t + 6}{t + 2} $. What is the simplified expression for $ S(t) $, and what is the smallest integer value of $ t $ for which $ S(t) $ is an integer?

["We begin by analyzing the function:", "$$\nS(t) = \frac{t^2 + 5t + 6}{t + 2}\n$$", "#### Step 1: Simplify the rational expression\nWe attempt to factor the numerator:", "$$\nt^2 + 5t + 6 = (t + 2)(t + 3)\n$$", "So the function becomes:", "$$\nS(t) = \frac{(t + 2)(t + 3)}{t + 2}\n$$", "For $ t <br/>\neq -2 $, we can cancel $ t + 2 $:", "$$\nS(t) = t + 3\n$$", "Thus, the simplified expression is $ S(t) = t + 3 $, for $ t <br/>\neq -2 $.", "#### Step 2: Find the smallest integer $ t $ such that $ S(t) $ is an integer\nSince $ S(t) = t + 3 $, and $ t $ is a real number (representing time in weeks), $ S(t) $ is an integer whenever $ t $ is an integer, except $ t = -2 $, where the function is undefined.", "To find the smallest integer $ t $ such that $ S(t) $ is an integer, we exclude $ t = -2 $, and take the smallest integer greater than $ -2 $, which is:", "$$\nt = -1\n$$", "Check:", "$$\nS(-1) = \frac{(-1)^2 + 5(-1) + 6}{-1 + 2} = \frac{1 - 5 + 6}{1} = 2\n$$", "Which is an integer.", "#### Final Answer:\nThe simplified expression is $ S(t) = t + 3 $, and the smallest integer value of $ t $ for which $ S(t) $ is an integer is $\boxed{-1}$."]