Answer
$${\text{circle}}$$
Work Step by Step
$$\eqalign{
& 2{x^2} - 8x + 2{y^2} + 20y = 12 \cr
& {\text{Factor}} \cr
& 2\left( {{x^2} - 4x} \right) + 2\left( {{y^2} + 10y} \right) = 12 \cr
& {\text{Divide both sides by 2}} \cr
& \left( {{x^2} - 4x} \right) + \left( {{y^2} + 10y} \right) = 6 \cr
& {\text{Complete the square and factor}} \cr
& \left( {{x^2} - 4x + 4} \right) + \left( {{y^2} + 10y + 25} \right) = 6 + 4 + 25 \cr
& {\left( {x - 2} \right)^2} + {\left( {y + 5} \right)^2} = 35 \cr
& {\left( {x - 2} \right)^2} + {\left( {y + 5} \right)^2} = {\left( {\sqrt {35} } \right)^2} \cr
& {\text{The equation is written in the form }}{\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2} \cr
& {\text{Therefore,}} \cr
& {\text{The graph of the equation is a circle}} \cr} $$
