Hello, lil_Dk!
This quadratic has complex roots; I assume you're cool with that . . .
\(\displaystyle \L4g^2\,+\,3g\,+\,1\:=\:0\)
I prefer to get rid of the leading coefficient first . . .
Divide by 4: \(\displaystyle \L\,g^2\,+\,\frac{3}{4}g\,+\,\frac{1}{4}\:=\:0\)
"Move" the constant term: \(\displaystyle \L\,g^2\,+\,\frac{3}{4}g\;=\;-\frac{1}{4}\)
Complete the square:
Take one-half of
43 and square:
(21⋅43)2=649
Add to both sides: \(\displaystyle \L\,g^2\,+\,\frac{3}{4}g\,+\,\frac{9}{64}\;=\;-\frac{1}{4}\,+\,\frac{9}{64}\)
Simplify: \(\displaystyle \L\,\left(g\,+\,\frac{3}{8}\right)^2\:=\: -\frac{7}{64}\)
Take square roots: \(\displaystyle \L\,g\,+\,\frac{3}{8}\:=\:\pm\sqrt{-\frac{7}{64}}\:=\:\pm\frac{i\sqrt{7}}{8}\)
Therefore: \(\displaystyle \L\,g\:=\:-\frac{3}{8}\,\pm\,\frac{i\sqrt{7}}{8}\:=\:\frac{-3\,\pm\,i\sqrt{7}}{8}\)