#### Answer

$x=-3\pm\sqrt{5}$

#### Work Step by Step

Multiplying both sides by $2$, the given quadratic equation, $
\dfrac{1}{2}x^2+3x+2=0
,$ is equivalent to
\begin{array}{l}\require{cancel}
x^2+6x+4=0
.\end{array}
Using $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ or the Quadratic Formula, the solutions of the quadratic equation above are
\begin{array}{l}\require{cancel}
x=\dfrac{-6\pm\sqrt{6^2-4(1)(4)}}{2(1)}
\\\\
x=\dfrac{-6\pm\sqrt{36-16}}{2}
\\\\
x=\dfrac{-6\pm\sqrt{20}}{2}
\\\\
x=\dfrac{-6\pm\sqrt{4\cdot5}}{2}
\\\\
x=\dfrac{-6\pm\sqrt{(2)^2\cdot5}}{2}
\\\\
x=\dfrac{-6\pm2\sqrt{5}}{2}
\\\\
x=\dfrac{2(-3\pm\sqrt{5})}{2}
\\\\
x=\dfrac{\cancel{2}(-3\pm\sqrt{5})}{\cancel{2}}
\\\\
x=-3\pm\sqrt{5}
.\end{array}