a) ab(a+b) - bc(b+c) + ac(a-c)
= a2b + ab2 - b2c - bc2 + a2c - ac2
= abc + a2b + ab2 - b2c - bc2 + a2c - ac2 - abc
= bc(a-c) + ab(a-c) + b2(a - c)+ ac(a -c)
= (a-c)(bc + ab + b2 + ac)
= (a-c)[ b (c+b) + a(c+b)]
= (a-c)(b+a)(c+b)
b) (a+b+c)3 - a3 - b3 - c3
= [(a+b) + c]3 - a3 - b3 - c3
= (a+b)3 + 3(a+b)2c + 3(a+b)c2 + c3 -a3 - b3 - c3
= a3 + 3a2b +3ab2 + b3 + 3(a2 + 2ab + b2) c + 3ac2 + 3bc2 - a3 - b3 - c3
=a3 + 3ab(a +b) + 3a2c + 6abc + 3b2c + 3ac2 + 3bc2 -a3 - b3 - c3
= 3ab(a+b) + 3a2c + 3abc + 3abc + 3b2c + 3ac2+ 3bc2
= 3ab(a+b) + 3ac(a +b) + 3bc(a + b) + 3c2 (a +b)
= (a+b) (3ab + 3ac + 3bc + 3c2)
= 3(a+b)(ab + ac + bc + c2)
=3(a+b)[a(b+c) + c(b+c)]
= 3(a+b)(a+c)(b+c)
c) a3 - b3 + c3 + 3abc
= a3 - (b3 - c3)+3abc
= a3 - [ (b-c) ( b2 + bc + c2)] + 3abc
= a3 - [(b-c) ( b2- 2bc + c2 + 3bc] + 3abc
=a3 - (b-c) [(b-c)2 + 3bc] + 3abc
=a3 - (b-c)3 - 3bc(b-c) + 3abc
=(a-b+c)[a2 - a(b-c) + (b-c)2] + 3bc ( a - b+c)
= (a-b+c) (a2 - ab + ac + b2 - 2bc + c2 + 3bc)
= (a-b+c)[a2 - ab + b2 + c2 + ac + bc)