分解因式的几道小题x^3-xy^2+x^2y-y^3 -a^2-b^2+2ab (ax+by)^2+(bx-ay)^2

x^3 - x y^2 + x^2 y - y^3
= (x^3 + x^2 y) - (x y^2 + y^3)
= x^2 (x + y) - y^2 (x + y)
= (x + y) (x^2 - y^2)
= (x - y) (x + y)^2
- a^2 - b^2 + 2 a b
= - (a^2 + b^2 - 2 a b)
= - (a - b)^2
(a x + b y)^2 + (b x - a y)^2
= a^2 x^2 + b^2 y^2 + 2 a x b y + b^2 x^2 + a^2 y^2 - 2 b x a y
= a^2 x^2 + a^2 y^2 + b^2 x^2 + b^2 y^2 + 2 a x b y - 2 b x a y
= a^2 (x^2 + y^2) + b^2 (x^2 + y^2)
= (a^2 + b^2) (x^2 + y^2)
x^3 + 3 x^2 + 3 x + 9
= x^2 (x + 3) + 3 (x + 3)
= (x + 3) (x^2 + 3)
x^3 y + 3 x - 2 x^2 y^2 - 6 y
= x^3 y- 2 x^2 y^2 + 3 x - 6 y
= x^2 y (x - 2 y) + 2 (x - 2 y)
= (x - 2 y) (x^2 y + 2)
3 a^2 + b c - 3 a c - a b
= (3 a^2 - 3 a c) - (a b - b c)
= 3 a (a - c) - b (a - c)
= (a - c) (3 a - b)
a^2 m + b n - a n - a b m
= (a^2 m - a b m) - (a n - b n)
= a m (a - b) - n (a - b)
= (a - b) (a m - n)
1 - m^2 - n^2 + 2 m n
= 1 - (m^2 + n^2 - 2 m n)
= 1 - (m - n)^2
= [1 + (m - n)] [1 - (m - n)]
= (1 + m - n) (1 - m + n)
x^4 y + 2 x^3 y^2 - x^2 y - 2 x y^2
= (x^4 y + 2 x^3 y^2) - (x^2 y + 2 x y^2)
= x^3 y (x + 2 y) - x y (x + 2 y)
= (x + 2 y) (x^3 y - x y)
= (x + 2 y) x y (x^2 - 1)
= x y (x + 1) (x - 1) (x + 2y)