Find the roots of the quadratic equation by applying the quadratic formula: $4 x^{2}+4 \sqrt{3} x+3=0$.

(D) Given quadratic equation: $4 x^{2}+4 \sqrt{3} x+3=0$.
Comparing this equation with the standard form $a x^{2}+b x+c=0$,we get $a=4, b=4 \sqrt{3}, c=3$.
The quadratic formula is given by $x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}$.
First,calculate the discriminant $D = b^{2}-4 a c = (4 \sqrt{3})^{2} - 4(4)(3) = 48 - 48 = 0$.
Since $D=0$,the roots are real and equal.
Substituting the values into the formula:
$x = \frac{-4 \sqrt{3} \pm \sqrt{0}}{2(4)}$
$x = \frac{-4 \sqrt{3}}{8}$
$x = \frac{-\sqrt{3}}{2}$.
Thus,the roots are $x = \frac{-\sqrt{3}}{2}, \frac{-\sqrt{3}}{2}$.