Factorize each of the following expressions:
(1) `x^2-2a x-2b a+b x`
(2) `x^3-2x^2y+3x y^2-6y^3`
(3)`a b x^2+(a y-b)x-y`
(4) `(a c+b y)^2+(b x-a y)^2`

Explanation

(i)`x^2-2a x-2b a+b x`
`=x(x-2a)+b(x-2a)`
`=(x+b)(x-2a)`

(ii) `x^3-2x^2y+3x y^2-6y^3`
`=x^2(x-2y)+3y(x-2y)`
`=(x^2+3y)(x-2y)`

(iii)`a b x^2+(a y-b)x-y`
`=abx^2+ayx-bx-y`
`=ax(bx+y)-1(bx+y)`
`=(ax-1)(bx+y)`

(iv) `(a c+b y)^2+(b x-a y)^2`
`=a^2x^2 +b^2y^2 +2abxy+b^2x^2+a^2y^2 −2abxy`
`=x^2(a^2+b^2)+y^2(a^2+b^2)`
`=(x^2+y^2)(a^2+b^2)`