a) x4 + 2x3 - 4x - 4
= (x4 - 4) + (2x3 - 4x)
= [(x2)2 - 22] + (2x3 - 4x)
= (x2 + 2)(x2 - 2) + 2x(x2 - 2)
= (x2 - 2)(x2 + 2 + 2x)
= (x - \(\sqrt{2}\))(x + \(\sqrt{2}\))(x2 + 2 + 2x)
b) bc(b + c) + ca(c - a) - ab(a + b)
= bc(b + c) + ac2 - a2c - a2b - ab2
= bc(b + c) - a2(b + c) - a(b + c)(b - c)
= (b + c)(bc - a2 - ab + ac)
= (b + c)[c(b + a) - a(a + b)]
= (b + c)(b + a)(c - a)
c) a5 - ax4 + a4x - x5
= (a5 - ax4) + (a4x - x5)
= a(a4 - x4) + x(a4 - x4)
= (a4 - x4)(a + x)
= [(a2)2 - (x2)2](a + x)
= (a2 - x2)(a2 + x2)(a + x)
= (a + x)(a - x)(a2 + x2)(a + x)
= (a + x)2(a - x)(a2 + x2)
d) (x2 + y2 - 5)2 - 4(xy + 2)2
= (x2 + y2 - 5)2 - (2xy + 4)2
= (x2 + y2 - 5 - 2xy - 4)(x2 + y2 - 5 + 2xy + 4)
= [(x - y)2 - 9][(x + y)2 - 1]
= [(x - y)2 - 32][(x + y)2 - 12]
= (x - y - 3)(x - y + 3)(x + y - 1)(x + y + 1)
