1.x(a-2)^n-y(2-a)^n
= x(a-2)^n-y(a-2)^n(-1)^n
= (x-(-1)^n * y)(a-2)^n
2.x^2-5xy+6y^2-3x+6y
= x^2-5xy+(25/4)y^2-(1/4)y^2-3x+6y
= (x-5y/2)^2-2(3/2)x+2(15/4)y-(3/2)y-(1/4)y^2
= (x-5y/2)^2-2(3/2)(x-5y/2)+(3/2)^2-(3/2)^2-(3/2)y-(1/4)y^2
= (x-5y/2-3/2)^2-(3/2)^2-2(y/2)(3/2)-(y/2)^2
= (x-5y/2-3/2)^2-(y/2+3/2)^2
= (x-5y/2-3/2-y/2-3/2)(x-5y/2-3/2+y/2+3/2)
= (x-3y-3)(x-2y)
3.125x^3-y^3
= (5x)^3 - y^3
= (5x-y)(25x^2+5xy+y^2)
4.a^4+1/4
= a^4+a^2+1/4-a^2
= (a^2+1/2)^2-a^2
= (a^2+a+1/2)(a^2-a+1/2)
= (1/4)(2a^2+2a+1)(2a^2-2a+1)