1. **State the problem:** Simplify the expression $$\frac{v^2 - 3v + 2}{v^2 - 1}$$ and find its simplified form. 2. **Recall the formula and rules:** To simplify a rational expression, factor both numerator and denominator and then cancel any common factors. 3. **Factor the numerator:** $$v^2 - 3v + 2 = (v - 1)(v - 2)$$ 4. **Factor the denominator:** $$v^2 - 1 = (v - 1)(v + 1)$$ 5. **Write the expression with factors:** $$\frac{(v - 1)(v - 2)}{(v - 1)(v + 1)}$$ 6. **Cancel the common factor $(v - 1)$:** $$\frac{\cancel{(v - 1)}(v - 2)}{\cancel{(v - 1)}(v + 1)} = \frac{v - 2}{v + 1}$$ 7. **Final simplified expression:** $$\boxed{\frac{v - 2}{v + 1}}$$ This is the simplified form of the original expression, valid for all $v \neq 1$ (since $v=1$ makes the original denominator zero).