2x^2 + 5x - 3 = 0

1. Identify the problem: The problem asks to solve the equation $2x^2 + 5x - 3 = 0$ for $x$.

1. Choose a method: We can solve this quadratic equation by factoring. We look for two numbers that multiply to $a \cdot c = 2 \cdot (-3) = -6$ and add to $b = 5$.

1. Find the factors: The numbers $6$ and $-1$ satisfy these conditions: $6 \cdot (-1) = -6$ and $6 + (-1) = 5$.

1. Rewrite the middle term: Split the middle term $5x$ using these factors: $2x^2 + 6x - x - 3 = 0$.

1. Factor by grouping: Group the terms: $(2x^2 + 6x) - (x + 3) = 0$. Factor out the common terms: $2x(x + 3) - 1(x + 3) = 0$.

1. Factor out the common binomial: Factor out $(x + 3)$: $(2x - 1)(x + 3) = 0$.

7. Solve for x: Set each factor to zero:
- $2x - 1 = 0 \implies 2x = 1 \implies x = \frac{1}{2}$
- $x + 3 = 0 \implies x = -3$