1. **State the problem:** Find the explicit form of the function $y$ from the equation $$\sqrt{3x^4 + 2x^2 + 2} + \ln\left(\frac{y^2}{4x}\right) + \ln\left(\frac{1}{y}\right) + \frac{1}{5x^2 + 1} = 5$$ where $x > 0$ and $y > 0$. 2. **Combine logarithmic terms:** Recall the logarithm property: $\ln(a) + \ln(b) = \ln(ab)$. Combine the two logarithms: $$\ln\left(\frac{y^2}{4x}\right) + \ln\left(\frac{1}{y}\right) = \ln\left(\frac{y^2}{4x} \times \frac{1}{y}\right) = \ln\left(\frac{y}{4x}\right)$$ 3. **Rewrite the equation:** $$\sqrt{3x^4 + 2x^2 + 2} + \ln\left(\frac{y}{4x}\right) + \frac{1}{5x^2 + 1} = 5$$ 4. **Isolate the logarithmic term:** $$\ln\left(\frac{y}{4x}\right) = 5 - \sqrt{3x^4 + 2x^2 + 2} - \frac{1}{5x^2 + 1}$$ 5. **Exponentiate both sides to remove the logarithm:** $$\frac{y}{4x} = e^{5 - \sqrt{3x^4 + 2x^2 + 2} - \frac{1}{5x^2 + 1}}$$ 6. **Solve for $y$ explicitly:** $$y = 4x \times e^{5 - \sqrt{3x^4 + 2x^2 + 2} - \frac{1}{5x^2 + 1}}$$ **Final explicit form:** $$\boxed{y = 4x e^{5 - \sqrt{3x^4 + 2x^2 + 2} - \frac{1}{5x^2 + 1}}}$$ This expression gives $y$ explicitly in terms of $x$ for $x > 0$ and $y > 0$.