1. **Stating the problem:** Simplify or analyze the rational function $$f(x) = \frac{x^2 + 2x + 4}{3x + 2}$$. 2. **Formula and rules:** A rational function is a ratio of two polynomials. Important rules include: - The denominator cannot be zero. - Simplify by factoring numerator and denominator if possible. 3. **Check the denominator:** Set denominator equal to zero to find restrictions: $$3x + 2 = 0 \implies x = -\frac{2}{3}$$ So, $$x \neq -\frac{2}{3}$$. 4. **Factor numerator if possible:** $$x^2 + 2x + 4$$ does not factor nicely over the reals (discriminant $\Delta = 2^2 - 4 \cdot 1 \cdot 4 = 4 - 16 = -12 < 0$), so it is prime. 5. **Conclusion:** The function cannot be simplified further. The domain is all real numbers except $$x = -\frac{2}{3}$$. **Final answer:** $$f(x) = \frac{x^2 + 2x + 4}{3x + 2}, \quad x \neq -\frac{2}{3}$$