4x^2+9y^2+8x-12y+12
=(4x^2+8x+4)+(9y^2-12y+4)+4
=(2x+2)^2+(3y-2)^2+4
(2x+2)^2≥0
(3y-2)^2≥0
(2x+2)^2+(3y-2)^2+4＞0
即：4x^2+9y^2+8x-12y+12＞0
所以：无论x,y取何值,4x^2+9y^2+8x-12y+12总是正数.