x2+2xy+3y2+2x+6y+4=
=(x2+2xy+y2)+(2y2+6y+2x+4)=
=(x+y)2+2(y2+2y+1)+2x+2y+2=
=(x+y)2+2(y+1)2+2(x+y)+2=
=((x+y)2+2(x+y)+1)+2(y+1)2+1=
=(x+y+1)2+2(y+1)2+1>0
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x2+2xy+3y2+2x+6y+4=
=(x2+2xy+y2)+(2y2+6y+2x+4)=
=(x+y)2+2(y2+2y+1)+2x+2y+2=
=(x+y)2+2(y+1)2+2(x+y)+2=
=((x+y)2+2(x+y)+1)+2(y+1)2+1=
=(x+y+1)2+2(y+1)2+1>0