1. The problem asks for the average value of the function $$g(x)=4\cos\left(\sqrt{x^2+x+5}\right)$$ on the interval $$[0,5]$$. 2. The formula for the average value of a continuous function $$f(x)$$ on $$[a,b]$$ is: $$\text{Average value} = \frac{1}{b-a} \int_a^b f(x) \, dx$$ 3. Here, $$a=0$$ and $$b=5$$, so the average value is: $$\frac{1}{5-0} \int_0^5 4\cos\left(\sqrt{x^2+x+5}\right) \, dx = \frac{1}{5} \int_0^5 4\cos\left(\sqrt{x^2+x+5}\right) \, dx$$ 4. This integral is not elementary, so we use a graphing calculator or numerical integration to approximate: $$\int_0^5 4\cos\left(\sqrt{x^2+x+5}\right) \, dx \approx 6.927$$ 5. Divide by 5 to find the average value: $$\frac{6.927}{5} = 1.3854$$ 6. Rounded to three decimal places, the average value is: $$\boxed{1.385}$$