1. 10a(x-y)^2-5b(y-x)=10a(x-y)^2+5b(x-y)=5(x-y)[2a(x-y)+b]=5(x-y)(2ax-2ay+b)
2.b^2+(b+1)^2+(b^2+b)^2=[b(b+1)]^2+b^2+b^2+2b+1=[b(b+1)]^2+2b^2+2b+1=[b(b+1)]^2+2[b(b+1)]+1=[b(b+1)+1]^2=(b^2+b+1)^2
3.(x+y)(x+y+2xy)+(xy+1)(xy-1)=(x+y)^2+2xy(x+y)+(xy)^2-1=[(x+y)+xy]^2-1=(x+y+xy+1)(x+y+xy-1)
4.a^2(b-c)+b^2(c-a)+c^2(a-b)=a^2(b-c)+b^2c-b^2a+c^2a-c^2b=a^2(b-c)+a(c^2-b^2)+bc(b-c)=a^2(b-c)+a(c-b)(c+b)+bc(b-c)=(b-c)(a^2-ac-ab+bc)=(b-c)(a^2-ac-ab+bc)=(b-c)[a(a-c)-b(a-c)]=(b-c)(a-c)(a-b)
5.(am+bn)^2+(bm-an)^2=a^2m^2+2abmn+b^2n^2+b^2m^2-2abmn+a^2n^2=a^2m^2+b^2n^2+b^2m^2+a^2n^2
=a^2(m^2+n^2)+b^2(m^2+n^2)=(a^2+b^2)(m^2+n^2)
6.(1-a^2)(1-b^2)-4ab=1-a^2-b^2+a^2b^2-2ab-2ab=(a^2b^2-2ab+1)-(a^2+2ab+b^2)=(ab-1)^2-(a+b)^2=(ab+a+b-1)(ab-a-b-1)
7.x^3+3x^2-4=x^3+2x^2+x^2-4=(x^3+2x^2)+(x^2-4)=x^2(x+2)+(x+2)(x-2)=(x+2)(x^2+x-2)=(x+2)(x+2)(x-1)=(x+2)^2(x-1)
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