1.x^2+2xy+4(x+1)+y^2+4y
=(x^2+2xy+y^2)+4x+4y+4
=(x+y)^2+4(x+y)+4
=(x+y+2)^2
2.(1-a^2)(1-b^2)-4ab
=1-a^2-b^2+a^2b^2-4ab
=(a^2b^2-2ab+1)-(a^2+2ab+b^2)
=(ab-1)^2-(a+b)^2
=(ab+a+b-1)(ab-a-b-1)
3.a^2(b-c)+b^2(c-a)+c^2(a-b)
=a^2b-a^2c+b^2c-b^2a+c^2a-c^2b
=ab(a-b)+bc(b-c)+ac(c-a)
=b(a^2-ab+cb-c^2)+ac(c-a)
=b[(a+c)(a-c)-b(a-c)]+ac(c-a)
=b(a-c)(a+b-c)-ac(a-c)
=(a-c)(ab+b^2-bc-ac)
=(a-c)(a-b)(b-c)
4.x^3+x^2+x-3
=(x^3-1)+(x^2-1)+(x-1)
=(x-1)(x^2+x+1)+(x-1)(x+1)+(x-1)
=(x-1)(x^2+x+1+x+1+1)
=(x-1)(x^2+2x+3)
5.a^3-4a^2-4a-5
=a^3-5a^2+a^2-4a-5
=a^2(a-5)+(a+1)(a-5)
=(a-5)(a^2+a+1)
6.a^3-6a^2+11a-6
=(a^3-1)-(6a^2-11a+5)
=(a-1)(a^2+a+1)-(6a-5)(a-1)
=(a-1)(a^2+a+1-6a+5)
=(a-1)(a^2-5a+6)
=(a-1)(a-2)(a-3)
7.(x+1)(x+2)(x+3)(x+4)-24
=[(x+1)(x+4)][(x+2)(x+3)]-24
=[(x^2+5x)+4][(x^2+5x)+6]-24
=(x^2+5x)^2+10(x^2+5x)+24-24
=(x^2+5x)(x^2+5x+10)
=x(x+5)(x^2+5x+10)