1. We need to evaluate $I=\int_1^2 e^x\,x^x\left(2+\ln x\right)\,dx.$

2. Observe the structure of the integrand. Let $f(x)=e^x x^x.$ Then differentiate: $f'(x)=e^x x^x+e^x\frac{d}{dx}(x^x).$

3. Recall $x^x=e^{x\ln x}$ so $\frac{d}{dx}(x^x)=x^x(\ln x+1).$ Hence, $f'(x)=e^x x^x+e^x x^x(\ln x+1)=e^x x^x(2+\ln x).$

4. Therefore the given integral is simply $I=\int_1^2 f'(x)\,dx=f(2)-f(1).$

5. Compute the endpoint values: $f(2)=e^2\cdot 2^2=4e^2,$ $f(1)=e^1\cdot 1^1=e.$ So, $I=4e^2-e=e(4e-1).$

6. Compare with the options:

• A: $e(4e+1)$
• B: $e(2e-1)$
• C: $e(4e-1)$
• D: $4e^2-1$

Thus the correct option is $\boxed{\text{C}}.$

7. Comparison with stored answer: stored correct answer is C, which matches our result.