1. **State the problem:** Solve the quadratic equation $$2x^2 - 5x + 3 = 0$$. 2. **Recall the factoring method:** To solve quadratic equations by factoring, we look for two binomials whose product equals the quadratic. 3. **Group terms:** Rewrite the middle term to help factor by grouping: $$2x^2 - 6x + x + 3 = 0$$ 4. **Factor by grouping:** $$ (2x^2 - 6x) + (x + 3) = 0 $$ $$ 2x(x - 3) + 1(x - 3) = 0 $$ 5. **Factor out the common binomial:** $$ (2x + 1)(x - 3) = 0 $$ 6. **Set each factor equal to zero and solve:** $$ 2x + 1 = 0 \implies 2x = -1 \implies x = \frac{-1}{2} $$ $$ x - 3 = 0 \implies x = 3 $$ 7. **Final answer:** $$ x = -\frac{1}{2}, 3 $$ This means the solutions to the quadratic equation are $x = -\frac{1}{2}$ and $x = 3$.