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Since the exponent function \(y=2^x\) is an increasing function, the smallest value of \(x^2+2x+3\) gives the smallest value of \(y=2^{x^2}+2^{x+3}\).

Since \(x^2+2x+3 = (x+1)^2 + 2\), the exponent \((x+1)^2 + 2\) has a minimum value when \(x = -1\).

Therefore, the answer is C.

Answer : C

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