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\begin{array}{ll} \text{Average rate of change} &=\dfrac{g(a)-g(0)}{a-0}  &\text{Evaluate.}\\ &=\dfrac{\left(a^{2}+3 a+1\right)-\left(0^{2}+3(0)+1\right)}{a-0} & \text { Simplify. } \\ &=\dfrac{a^{2}+3 a+1-1}{a} & \text { Simplify and factor. } \\ &=\dfrac{a(a+3)}{a} & \text { Divide by the common factor } a . \\ &=a+3 & \end{array}