1. **State the problem:** Simplify the expression $$\frac{2x^2 - 5x + 3}{2x^2 - x - 3}$$. 2. **Factor numerator and denominator:** - Numerator: $2x^2 - 5x + 3$ Find two numbers that multiply to $2 \times 3 = 6$ and add to $-5$: these are $-2$ and $-3$. Rewrite as $2x^2 - 2x - 3x + 3$. Factor by grouping: $$2x(x - 1) - 3(x - 1) = (2x - 3)(x - 1)$$. - Denominator: $2x^2 - x - 3$ Find two numbers that multiply to $2 \times (-3) = -6$ and add to $-1$: these are $-3$ and $2$. Rewrite as $2x^2 - 3x + 2x - 3$. Factor by grouping: $$x(2x - 3) + 1(2x - 3) = (x + 1)(2x - 3)$$. 3. **Rewrite the expression:** $$\frac{(2x - 3)(x - 1)}{(x + 1)(2x - 3)}$$ 4. **Cancel common factors:** $$\frac{\cancel{(2x - 3)}(x - 1)}{(x + 1)\cancel{(2x - 3)}} = \frac{x - 1}{x + 1}$$ 5. **Final answer:** $$\boxed{\frac{x - 1}{x + 1}}$$ This simplification is valid as long as $2x - 3 \neq 0$, i.e., $x \neq \frac{3}{2}$, and $x + 1 \neq 0$, i.e., $x \neq -1$ (to avoid division by zero).