1. **State the problem:** We are given the derivative of a function $g(x)$ as $$\frac{dg}{dx} = \frac{3}{\sqrt{x}} \cdot \left(1 - 2x^2\right)$$ and the initial condition $g(8) = 12$. We need to find the original function $g(x)$.\n\n2. **Recall the formula:** To find $g(x)$ from its derivative, we integrate the derivative: $$g(x) = \int \frac{dg}{dx} \, dx + C$$ where $C$ is the constant of integration.\n\n3. **Rewrite the derivative for easier integration:** Note that $\frac{3}{\sqrt{x}} = 3x^{-\frac{1}{2}}$, so \n$$\frac{dg}{dx} = 3x^{-\frac{1}{2}} (1 - 2x^2) = 3x^{-\frac{1}{2}} - 6x^{\frac{3}{2}}$$\n\n4. **Integrate term-by-term:**\n$$g(x) = \int \left(3x^{-\frac{1}{2}} - 6x^{\frac{3}{2}}\right) dx + C = 3 \int x^{-\frac{1}{2}} dx - 6 \int x^{\frac{3}{2}} dx + C$$\n\n5. **Use the power rule for integration:** For $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (valid for $n \neq -1$).\n- For $\int x^{-\frac{1}{2}} dx$, $n = -\frac{1}{2}$, so\n$$\int x^{-\frac{1}{2}} dx = \frac{x^{\frac{1}{2}}}{\frac{1}{2}} = 2x^{\frac{1}{2}}$$\n- For $\int x^{\frac{3}{2}} dx$, $n = \frac{3}{2}$, so\n$$\int x^{\frac{3}{2}} dx = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} = \frac{2}{5} x^{\frac{5}{2}}$$\n\n6. **Substitute back:**\n$$g(x) = 3 \cdot 2x^{\frac{1}{2}} - 6 \cdot \frac{2}{5} x^{\frac{5}{2}} + C = 6x^{\frac{1}{2}} - \frac{12}{5} x^{\frac{5}{2}} + C$$\n\n7. **Apply the initial condition $g(8) = 12$ to find $C$:**\nCalculate $g(8)$:\n$$g(8) = 6 \sqrt{8} - \frac{12}{5} (8)^{\frac{5}{2}} + C = 6 \cdot 2\sqrt{2} - \frac{12}{5} \cdot 8^{2} \cdot \sqrt{8} + C$$\nSince $\sqrt{8} = 2\sqrt{2}$ and $8^2 = 64$,\n$$g(8) = 6 \cdot 2\sqrt{2} - \frac{12}{5} \cdot 64 \cdot 2\sqrt{2} + C = 12\sqrt{2} - \frac{12}{5} \cdot 128 \sqrt{2} + C$$\nCalculate the coefficient:\n$$\frac{12}{5} \cdot 128 = \frac{12 \times 128}{5} = \frac{1536}{5}$$\nSo,\n$$g(8) = 12\sqrt{2} - \frac{1536}{5} \sqrt{2} + C = \left(12 - \frac{1536}{5}\right) \sqrt{2} + C$$\nSimplify inside the parentheses:\n$$12 = \frac{60}{5}$$\n$$\frac{60}{5} - \frac{1536}{5} = -\frac{1476}{5}$$\nSo,\n$$g(8) = -\frac{1476}{5} \sqrt{2} + C = 12$$\n\n8. **Solve for $C$:**\n$$C = 12 + \frac{1476}{5} \sqrt{2}$$\n\n9. **Final answer:**\n$$g(x) = 6x^{\frac{1}{2}} - \frac{12}{5} x^{\frac{5}{2}} + 12 + \frac{1476}{5} \sqrt{2}$$