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A herpetologist is modeling the population growth of a species of endangered reptiles using a quadratic function \( P(t) = at^2 + bt + c \), where \( t \) is time in years. If \( P(t) \) reaches its maximum at \( t = 5 \) years and \( a < 0 \), determine the relationship between \( a, b, \) and \( c \).
The vertex form of a quadratic function \( P(t) = at^2 + bt + c \) gives the time at which the maximum value occurs as \( t = - rac{b}{2a} \). Given that the maximum occurs at \( t = 5 \), we have:

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