Solve this

Since, $x=\frac{-1}{2}$ is a root of the quadratic equation $3 x^{2}+2 k x+3=0$,

then, it must satisfies the equation.

$3\left(-\frac{1}{2}\right)^{2}+2 k\left(-\frac{1}{2}\right)+3=0$

$\Rightarrow 3\left(\frac{1}{4}\right)-k+3=0$

$\Rightarrow \frac{3}{4}-k+3=0$

$\Rightarrow \frac{3-4 k+12}{4}=0$

$\Rightarrow 3-4 k+12=0$

$\Rightarrow 4 k=15$

$\Rightarrow k=\frac{15}{4}$

Hence, the value of $k$ is $\frac{15}{4}$.