1. **State the problem:** We want to find the derivative of the function $$f(x) = e^{x^2 + 3x}$$. 2. **Recall the formula:** The derivative of an exponential function with base $e$ is given by $$\frac{d}{dx} e^{g(x)} = e^{g(x)} \cdot g'(x)$$ where $g(x)$ is the exponent function. 3. **Identify the inner function:** Here, $$g(x) = x^2 + 3x$$. 4. **Find the derivative of the inner function:** $$g'(x) = \frac{d}{dx}(x^2 + 3x) = 2x + 3$$. 5. **Apply the chain rule:** $$\frac{d}{dx} e^{x^2 + 3x} = e^{x^2 + 3x} \cdot (2x + 3)$$. 6. **Final answer:** $$\boxed{\frac{d}{dx} e^{x^2 + 3x} = e^{x^2 + 3x} (2x + 3)}$$