Given:

#color(white)("XXX")x^2+4y^2-8x+16y+16=0#

Group #x# and #y# terms (and constant) separately as

#color(white)("XXX")color(blue)(x^2-8x)+color(red)(4(y^2+4y))+color(green)(16)=0#

Complete the squares for both the #x# and the #y# sub-expressions:

#color(white)("XXX")color(blue)(x^2-8x+16) + color(red)(4(y^2+4y+4))+color(green)(16)-color(blue)(16)-color(red)(4(4))=0#

Reduce #x# and #y# sub-expressions to squared binomials and move constant to the right side:

#color(white)("XXX")(x-4)^2+4(y+2)^2= 16#

or

#color(white)("XXX")(x-4)^2+4(y+2)^2= 4^2#

Divide both sides by #(4^2)#

and replace #(y+2)# with #(y-(-2))#

#color(white)("XXX")((x-4)^2)/(4^2)+(y-(-2))^2/(2^2)=1#

Note that the general form of an ellipse is

#color(white)("XXX")(x-h)^2/(a^2)+((y-k)^2)/(b^2)=1#

with

#color(white)("XXX")#center at #(h,k)#,

#color(white)("XXX")#x-axis radius: #(a)#, and

#color(white)("XXX")#y-axis radius: #(b)#

graph{x^2+4y^2-8x+16y+16=0 [-2.195, 10.29, -4.895, 1.345]}