Answer
$x=-1-i\sqrt{2}
\text{ OR }
x=-1+i\sqrt{2}
$
Work Step by Step
In the given equation,
\begin{align*}
x^2+2x+3=0
,\end{align*} $a=
1
,$ $b=
2
,$ and $c=
3
.$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then
\begin{align*}\require{cancel}x&=
\dfrac{-2\pm\sqrt{2^2-4(1)(3)}}{2(1)}
\\\\&=
\dfrac{-2\pm\sqrt{4-12}}{2}
\\\\&=
\dfrac{-2\pm\sqrt{-8}}{2}
\\\\&=
\dfrac{-2\pm\sqrt{-1}\cdot\sqrt{4}\cdot\sqrt{2}}{2}
\\\\&=
\dfrac{-2\pm\sqrt{-1}\cdot2\cdot\sqrt{2}}{2}
\\\\&=
\dfrac{-2\pm i\cdot2\cdot\sqrt{2}}{2}
&\text{ (use $i=\sqrt{-1}$)}
\\\\&=
\dfrac{-2\pm 2i\sqrt{2}}{2}
\\\\&=
\dfrac{-\cancel2^1\pm \cancel2^1i\sqrt{2}}{\cancel2^1}
&\text{ (divide by $2$)}
\\\\&=
-1\pm i\sqrt{2}
.\end{align*}
The solutions are $
x=-1-i\sqrt{2}
\text{ and }
x=-1+i\sqrt{2}
.$
