1. The problem is to simplify the function $$F(x) = \frac{\sqrt{x^2 + 3 - 2}}{x - 1}, \quad x \neq 1.$$\n\n2. First, simplify the expression inside the square root: $$x^2 + 3 - 2 = x^2 + 1.$$\n\n3. So the function becomes $$F(x) = \frac{\sqrt{x^2 + 1}}{x - 1}.$$\n\n4. Note that $$\sqrt{x^2 + 1}$$ cannot be simplified further because $$x^2 + 1$$ is always positive and does not factor nicely.\n\n5. The domain excludes $$x = 1$$ because the denominator would be zero, which is undefined.\n\n6. Therefore, the simplified function is $$F(x) = \frac{\sqrt{x^2 + 1}}{x - 1}, \quad x \neq 1.$$