The problem asks us to find the quadratic equation given the sum and product of its roots. The sum of the roots is $-2$ and the product of the roots is $-\frac{3}{2}$.

1. Problem Description

The problem asks us to find the quadratic equation given the sum and product of its roots. The sum of the roots is $-2$ and the product of the roots is $-\frac{3}{2}$.

2. Solution Steps

A quadratic equation can be expressed in terms of the sum and product of its roots as follows:

$x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$

We are given that the sum of the roots is $-2$ and the product of the roots is $-\frac{3}{2}$. Substituting these values into the formula, we get:

$x^2 - (-2)x + (-\frac{3}{2}) = 0$

$x^2 + 2x - \frac{3}{2} = 0$

To eliminate the fraction, we can multiply the entire equation by 2:

$2(x^2 + 2x - \frac{3}{2}) = 2(0)$

$2x^2 + 4x - 3 = 0$

3. Final Answer

The quadratic equation is $2x^2 + 4x - 3 = 0$.