Complete the square to describe the conics in Exercises $18-21$.

$2x^2+4x+y^2-6y-3=0$

We have given the following equation :-

$2x^2+4x+y^2-6y-3=0$

We have to complete the squares and also describes the conic sections after reducing the given equation.

The conic section will be easily determine after reducing the given equation to complete squares.

The given equation is :-

$2x^2+4x+y^2-6y-3=0$

To reduce it to complete squares add $2$and $9$on both sides, then we have :-

$2x^2+4x+2+y^2-6y+9-3=2+9$

Simplify the equation as following :-

$$\begin{gathered} \left(2x^2+4x+2\right)+\left(y^2-6y+9\right)-3=11 \\ \Rightarrow 2\left(x^2+2x+1\right)+\left(y^2-6y+9\right)=11+3 \\ \Rightarrow 2{\left(x+1\right)}^2+{\left(y-3\right)}^2=14 \\ \Rightarrow \frac{2{\left(x+1\right)}^2}{14}+\frac{{\left(y-3\right)}^2}{14}=1 \\ \Rightarrow \frac{{\left(x+1\right)}^2}{7}+\frac{{\left(y-3\right)}^2}{14}=1 \end{gathered}$$

This is the required equation/

The given equation reduced to following equation by completing the squares :-

$\frac{{\left(x+1\right)}^2}{7}+\frac{{\left(y-3\right)}^2}{14}=1$

We know that the general equation of the ellipse is :-

$\frac{{(x-h)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$

By comparing these two equations, we have graph of the given equation is ellipse center at $\left(-1,3\right)$and major axis is $y$-axis.