x^6+6x^5+15x^4+20x^3+15x^2+6x+1
=x^4(x^2+2x+1)+4x^3(x^2+2x+1)+6x^2(x^2+2x+1)+4x(x^2+2x+1)+(x^2+2x+1)
=x^4(x+1)^2+4x^3(x+1)^2+6x^2(x+1)^2+4x(x+1)^2+(x+1)^2
=(x+1)^2(x^4+4x^3+6x^2+4x+1)
=(x+1)^2[x^2(x^2+2x+1)+2x(x^2+2x+1)+(x^2+2x+1)]
=(x+1)^2[x^2(x+1)^2+2x(x+1)^2+(x+1)^2]
=(x+1)^4(x^2+2x+1)
=(x+1)^6