The problem is to factorize the quadratic expression $2x^2 + 5x - 3$ completely.

1. Problem Description

The problem is to factorize the quadratic expression $2x^2 + 5x - 3$ completely.

2. Solution Steps

We need to factorize the quadratic expression $2x^2 + 5x - 3$. We are looking for two binomials of the form $(ax + b)(cx + d)$ such that when multiplied, they result in the given quadratic.

We are looking for two numbers that multiply to $(2)(-3) = -6$ and add up to $5$. These two numbers are $6$ and $-1$.

We can rewrite the middle term $5x$ as $6x - x$.

$2x^2 + 5x - 3 = 2x^2 + 6x - x - 3$

Now, we factor by grouping:

$2x^2 + 6x - x - 3 = 2x(x + 3) - 1(x + 3)$

We can factor out the common term $(x+3)$:

$2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3)$

3. Final Answer

The factored form of the equation $2x^2 + 5x - 3$ is $(2x - 1)(x + 3)$.