1. **State the problem:** Find the derivative $f'(x)$ of the function $$f(x) = x^2 + 4x + \sqrt{x^3}.$$\n\n2. **Rewrite the function:** Recall that $$\sqrt{x^3} = x^{3/2}.$$ So, $$f(x) = x^2 + 4x + x^{3/2}.$$\n\n3. **Recall derivative rules:**\n- The derivative of $x^n$ is $$\frac{d}{dx} x^n = n x^{n-1}.$$\n- The derivative of a sum is the sum of derivatives.\n\n4. **Differentiate each term:**\n- $$\frac{d}{dx} x^2 = 2x,$$\n- $$\frac{d}{dx} 4x = 4,$$\n- $$\frac{d}{dx} x^{3/2} = \frac{3}{2} x^{\frac{3}{2} - 1} = \frac{3}{2} x^{1/2}.$$\n\n5. **Combine the derivatives:**\n$$f'(x) = 2x + 4 + \frac{3}{2} x^{1/2}.$$\n\n6. **Check given options:**\n- Option a: $2x + 4 + 32x^{1/2}$ (incorrect coefficient 32 instead of $\frac{3}{2}$)\n- Option b: $2x + 4$ (missing last term)\n- Option c: $x^2 + 32x^{1/2}$ (incorrect derivative and terms)\n- Option d: $2x + 12x^{1/2}$ (incorrect coefficient 12 instead of $\frac{3}{2}$)\n- Option e: None of the given options is correct.\n\n**Final answer:** e. None of the given options is correct.