1. **State the problem:** We are given a differentiable function $h$ with $h(-5)=10$ and its derivative $h'(x)=2-\sqrt{e^x+2x^2}$. We need to find $h(1)$. 2. **Formula used:** Since $h'(x)$ is the derivative of $h(x)$, we can find $h(1)$ by integrating $h'(x)$ from $-5$ to $1$ and adding $h(-5)$: $$h(1) = h(-5) + \int_{-5}^1 h'(x) \, dx$$ 3. **Set up the integral:** $$h(1) = 10 + \int_{-5}^1 \left(2 - \sqrt{e^x + 2x^2}\right) dx$$ 4. **Evaluate the integral:** This integral does not have a simple closed form, so we use a graphing calculator or numerical integration method to approximate: $$\int_{-5}^1 \left(2 - \sqrt{e^x + 2x^2}\right) dx \approx -3.591$$ 5. **Calculate $h(1)$:** $$h(1) = 10 + (-3.591) = 6.409$$ 6. **Final answer:** $$\boxed{6.409}$$ This is the value of $h(1)$ rounded to three decimal places.