1. **Stating the problem:** Simplify the function $$f(x) = \frac{2x^{2} + 3x - 1}{x + 2}$$ and analyze its behavior. 2. **Formula and rules:** For rational functions, simplify by factoring numerator and denominator and cancel common factors if possible. Remember, division by zero is undefined, so exclude values that make the denominator zero. 3. **Factor numerator:** $$2x^{2} + 3x - 1 = (2x - 1)(x + 1)$$ 4. **Rewrite function:** $$f(x) = \frac{(2x - 1)(x + 1)}{x + 2}$$ 5. **Check for common factors:** No common factors between numerator and denominator, so no simplification by cancellation. 6. **Domain:** Denominator cannot be zero, so $$x + 2 \neq 0 \Rightarrow x \neq -2$$ 7. **Final simplified form:** $$f(x) = \frac{(2x - 1)(x + 1)}{x + 2}$$ This is the simplest form. The function is undefined at $$x = -2$$. **Answer:** $$f(x) = \frac{(2x - 1)(x + 1)}{x + 2}, \quad x \neq -2$$