${x}^{2}+2x=2+y$
Use the commutative property to reorder the terms${x}^{2}+2x=y+2$
To complete the square, the same value needs to be added to both sides${x}^{2}+2x+?=y+2+?$
To complete the square ${x}^{2}+2x+1={\left( x+1 \right)}^{2}$ add $1$ to the expression${x}^{2}+2x+1=y+2+?$
Since $1$ was added to the left-hand side, also add $1$ to the right-hand side${x}^{2}+2x+1=y+2+1$
Use ${a}^{2}+2ab+{b}^{2}={\left( a+b \right)}^{2}$ to factor the expression${\left( x+1 \right)}^{2}=y+2+1$
Simplify the expression${\left( x+1 \right)}^{2}=y+3$
The equation can be written in the form ${\left( x-h \right)}^{2}=4p \times \left( y-k \right)$, so it represents a parabola with the vertex $\left( -1, -3\right)$$\textnormal{}\\textnormal{}t\textnormal{}e\textnormal{}x\textnormal{}t\textnormal{}n\textnormal{}o\textnormal{}r\textnormal{}m\textnormal{}a\textnormal{}l\textnormal{}{\textnormal{}P\textnormal{}a\textnormal{}r\textnormal{}a\textnormal{}b\textnormal{}o\textnormal{}l\textnormal{}a\textnormal{} \textnormal{}w\textnormal{}i\textnormal{}t\textnormal{}h\textnormal{} \textnormal{}t\textnormal{}h\textnormal{}e\textnormal{} \textnormal{}v\textnormal{}e\textnormal{}r\textnormal{}t\textnormal{}e\textnormal{}x\textnormal{} \textnormal{}}\textnormal{}A\textnormal{}R\textnormal{}G\textnormal{}1\textnormal{}$