(1) 3x^2+6x+3=3(x^2+2x+1)=3(x+1)^2.
(2) ax^2--4ax+4a=a(x^2--4x+4)=a(x--2)^2.
(3) xy^2--2x^2y^2+x^3y=xy(y-2xy+x^2).
(4) (x+y)^2+6(x+y)+9=[(x+y)+3]^2=(x+y+3)^2.
(5)16--24(x--y)+9(x--y)^2=[4--3(x--y)]^2=(4--3x+3y)^2.
(6) x^4y^4--8x^2y^2+16=(x^2y^2--4)^2=[(xy+2)(xy--2)]^2=(xy+2)^2(xy--2)^2.