x^4+x^3+2x^2+x+1
=x^4+x^3+x^2+x^2+x+1
=x^2(x^2+x+1)+x^2+x+1
=(x^2+1)(x^2+x+1)
-1-2x-x^2+y^2
=y^2-(1+2x+x^2)
=y^2-(x+1)^2
=(y+x+1)(y-x-1)
x^2+ax^2+x+ax-1-a
=x^2(a+1)+x(a+1)-(a+1)
=(a+1)(x^2+x-1)
x^3-2x^2-x+2+x^5-2x^4
=x^5-2x^4+x^3-2x^2-x+2
=x^4(x-2)+x^2(x-2)-(x-2)
=(x-2)(x^4+x^2-1)
x^3+x^2-y^3-y^2
=x^3-y^3+x^2-y^2
=(x-y)(x^2+x+1)+(x+y)(x-y)
=(x-y)(x^2+x+1+x+y)
=(x-y)(x^2+2x+y+1)
ab(c^2-d^2)-cd(a^2-b^2)
=abc^2-abd^2-a^2cd+b^2cd
=(abc^2-a^2cd)-abd^2+b^2cd
=ac(bc-ad)+bd(bc-ad)
=(bc-ad)(ac+bd)
x^8-y^8
=(x^4+y^4)(x^4-y^4)
=(x^4+y^4)(x^2+y^2)(x^2-y^2)
=(x^4+y^4)(x^2+y^2)(x+y)(x-y)
2(x+y)+6(x+y)^2-4(x+3)^3
=2(x+y)[1+3(x+y)-4(x+y)^2]
=2(x+y)[1-4(x+y)][1+(x+y)]
=2(x+y)(1-4x-4y)(1+x+y)
-(3a^2-5b^2)^2+(5a^2-3b^2)^2
=(5a^2-3b^2)^2-(3a^2-5b^2)^2
=[(5a^2-3b^2)-(3a^2-5b^2)][(5a^2-3b^2)+(3a^2-5b^2)]
=(2a^2+2b^2)(8a^2-8b^2)
=16(a^2+b^2)(a+b)(a-b)
(5x^2+2x-3)^2-(x^2-2x-3)^2
=(5x^2+2x-3)^2-(x^2-2x-3)^2
=[(5x^2+2x-3)-(x^2-2x-3)][(5x^2+2x-3)+(x^2-2x-3)]
=(6x^2+4x)(6x^2-6)
=12x(3x+2)(x+1)(x-1)
x^2(a+b)^2-2xy(a^2-b^2)+y^2(a-b)^2
=[x(a+b)-y(a-b)]^2
9a^2+x^2n+6a+2x^n+6ax^n+1
=x^2n+6ax^n+9a^2+6a+2x^n+1
=(x^n+3a)^2+2(x^n+3)+1
=(x^n+3a+1)^2