a.)
x^4 + 1
= (x^2)^2 + 1
= x^4 + 2x^2 +1
b.)
x^2 + 2xy + y^2 - x - y -12
= (x + y)^2 - (x + y) -12
= (x + y)(x + y - 1) - 12
c.)
(x^2 + x)^2 - 2(x^2 + 1) + 1
= x^2(x + 1)^2 - 2(x^2 + 1) + 1
= x^2(x^2 + 2x + 1) - 2(x^2 + 1) + 1
= x^4 + 2x^3 + x^2 - 2x^2 - 2 + 1
= x^4 + 2x^3 - x^2 -1
= x^3(x + 2) - (x^2 + 1)
d.)
4(2x - 3)^2 - 25(x + 7)^2
= (2(2x - 3))^2 - (5(x + 7))^2
= ((2(2x - 3)) - (5(x + 7))).((2(2x - 3)) + (5(x + 7)))
= (-x - 41).(9x + 29)