If the solution curve of the differential equation \(\frac{x + y - 2}{x + y - 1}\,\frac{dy}{dx} = \frac{x + y + 2}{x + y + 1}\), \(x + y > 2\), passes through the points \((\sqrt{2}, \sqrt{2})\) and \((2, \alpha)\), then \(2\alpha - \log_e\left(\frac{\alpha^2 + 4\alpha + 2}{6}\right)\) is equal to ________.