The sum and product of the zeros of a quadratic polynomial are

Let sum of quadratic polynomial is $\alpha+\beta=\frac{-1}{2}$

Product of the quadratic polynomial is $\alpha \beta=-3$

Let $S$ and $P$ denote the sum and product of the zeros of a polynomial as $\frac{-1}{2}$ and $-3$.

Then

The required polynomial $g(x)$ is given by

$g(x)=k\left(x^{2}-S x+P\right)$

$=k\left[x^{2}-\left(\frac{-1}{2}\right) x+(-3)\right]$

$=k\left[x^{2}+\frac{1}{2} x-3\right]$

Hence, the quadratic polynomial is $g(x)=k\left(x^{2}+\frac{x}{2}-3\right)$, where $k$ is any non-zero real number