Answer
$${\text{The equation describes an ellipse}}{\text{.}}$$
Work Step by Step
$$\eqalign{
& 4{x^2} + 8{y^2} = 16 \cr
& \frac{{4{x^2}}}{{16}} + \frac{{8{y^2}}}{{16}} = \frac{{16}}{{16}} \cr
& \frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1 \cr
& \left( a \right) \cr
& {\text{The equation is written in the form }}\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,{\text{ }}a > b > 0 \cr
& {\text{then, the equation describes an ellipse}}{\text{.}} \cr
& \underbrace {\frac{{{x^2}}}{4} + \frac{{{y^2}}}{2} = 1}_{\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1} \to a = 2,{\text{ }}b = \sqrt 2 \cr
& c = \sqrt {{a^2} - {b^2}} = \sqrt {4 - 2} = \sqrt 2 \cr
& \cr
& \left( b \right) \cr
& {\text{Foci: }}\left( { - c,0} \right){\text{ and }}\left( {c,0} \right) \cr
& {\text{Foci: }}\left( { - \sqrt 2 ,0} \right){\text{ and }}\left( {\sqrt 2 ,0} \right) \cr
& {\text{Vertices: }}\left( { - a,0} \right){\text{ and }}\left( {a,0} \right) \cr
& {\text{Vertices: }}\left( { - 2,0} \right){\text{ and }}\left( {2,0} \right) \cr
& \cr
& \left( c \right) \cr
& {\text{Eccentricity }} \cr
& e = \frac{c}{a} \cr
& e = \frac{{\sqrt 2 }}{2} \cr
& \cr
& \left( d \right) \cr
& {\text{Graph}} \cr} $$
