X (x-2) - y (y-2)

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(x - y) (y + x - 2)

Step-by-step explanation:

Factor the following:

x (x - 2) - y (y - 2)

x (x - 2) = x^2 - 2 x:

x^2 - 2 x - y (y - 2)

-y (y - 2) = 2 y - y^2:

-2 x + x^2 + 2 y - y^2

-2 x + x^2 + 2 y - y^2 = (x - 1) ^2 - (y - 1) ^2:

(x - 1) ^2 - (y - 1) ^2

Factor the difference of two squares. (x - 1) ^2 - (y - 1) ^2 = ((x - 1) - (y - 1)) ((x - 1) + (y - 1)):

(-1 + x - (y - 1)) (-1 + x - 1 + y)

Grouping like terms, - 1 + x - 1 + y = y + x + (-1 - 1):

y + x + (-1 - 1) (-1 + x - (y - 1))

-1 - 1 = - 2:

(-1 + x - (y - 1)) (y + x + - 2)

- (y - 1) = 1 - y:

(-1 + x + 1 - y) (y + x - 2)

Grouping like terms, - 1 + x + 1 - y = - y + x + (1 - 1):

-y + x + (1 - 1) (y + x - 2)

1 - 1 = 0:

Answer: (x - y) (y + x - 2)