`a)`
`x^2 - 4x^2y^2 + y^2 + 2xy`
`= x^2 + 2xy + y^2 - 4x^2y^2`
`= (x + y)^2 - (2xy)^2`
`= (x + y - 2xy)(x + y + 2xy)`
`b)`
`x^6 - y^6`
`(x^3)^2 - (y^3)^2`
`= (x^3 - y^3)(x^3 + y^3)`
`= (x - y)(x^2 + xy + y^2)(x + y)(x^2 - xy + y^2)`
`c)`
`(a + b + c)^2 + (a + b - c)^2 - 4c^2`
`= a^2 + b^2 + c^2 + 2ab + 2bc + 2ca + a^2 + b^2 + c^2 + 2ab - 2bc - 2ca - 4c^2`
`= 2a^2 + 2b^2 + 2c^2 + 4ab - 4c^2`
`= 2(a^2 + b^2 + c^2 + 2ab - 2c^2)`
`= 2(a^2 + b^2 + 2ab - c^2)`
`= 2[(a + b)^2 - c^2]`
`= 2(a + b + c)(a + b - c)`
