=a, a(b2+c2)+b(a2+c2)+c(a2+b2)+2abc

= ab2+ac2+ba2+bc2+ca2+cb2+2abc

= c2(a+b)+ab(a+b)+c(a2+b2+2ab)

= c2(a+b)+ab(a+b)+c(a+b)2

= (a+b)\(\left[c^2+ab+c\left(a+b\right)\right]\)

= (a+b)(c2+ab+ca+cb)

= (a+b)\(\left[c\left(a+c\right)+b\left(a+c\right)\right]\)

=(a+b)(a+c)(b+c)

b, a(b-c)3+b(c-a)3+c(a-b)3

= a(b-c)3-b\(\left[\left(b-c\right)+\left(a-b\right)\right]\)3+c(a-b)3

= a(b-c)3-b(b-c)3-3b(b-c)2(a-b)-3b(b-c)(a-b)2-b(a-b)3+c(a-b)3

= a(b-c)3-b(b-c)3-3b(b-c)(a-b)(b-c+a-b)-b(a-b)3+c(a-b)3

= a(b-c)3-b(b-c)3-3b(b-c)(a-b)(a-c)-b(a-b)3+c(a-b)3

= (b-c)3(a-b)-3b(b-c)(a-b)(a-c)-(a-b)3(b-c)

= (b-c)(a-b)\(\left[\left(b-c\right)^2-3b\left(a-c\right)-\left(a-b\right)^2\right]\)

=(b-c)(a-b)(b2-2bc+c2-3ab+3bc-a2+2ab-b2)

= (b-c)(a-b)(c2-a2+bc-ab)

= (b-c)(a-b)\(\left[\left(c-a\right)\left(c+a\right)+b\left(c-a\right)\right]\)

= (b-c)(a-b)(c-a)(c+a+b)

c, a2b2(a-b)+b2c2(b-c)+c2a2(c-a)

= a2b2(a-b)-b2c2\(\left[\left(a-b\right)+\left(c-a\right)\right]\)+c2a2(c-a)

= a2b2(a-b)-b2c2(a-b)-b2c2(c-a)+c2a2(c-a)

= b2(a-b)(a2-c2)+c2(c-a)(a2-b2)

= b2(a-b)(a-c)(a+c)-c2(a-c)(a-b)(a+b)

= (a-c)(a-b)\(\left[b^2\left(a+c\right)-c^2\left(a+b\right)\right]\)

= (a-c)(a-b)(b2a+b2c-c2a-c2b)

= (a-c)(a-b)\(\left[a\left(b^2-c^2\right)+bc\left(b-c\right)\right]\)

= (a-c)(a-b)\(\left[a\left(b-c\right)\left(b+c\right)+bc\left(b-c\right)\right]\)

= (a-c)(a-b)(b-c)\(\left[a\left(b+c\right)+bc\right]\)

= (a-c)(a-b)(b-c)(ab+ac+bc)

d, a4(b-c)+b4(c-a)+c4(a-b)

= a4(b-c)-b4[(b-c)+(a-b)]+c4(a-b)
= (b-c)(a4-b4)+(a-b)(c4-b4)
= (b-c)(a2-b2)(a2+b2)+(a-b)(c2-b2)(c2+b2)
= (b-c)(a-b)(a+b)(a^2+b^2)-(a-b)(b-c)(b+c)(b2+c2)
= (b-c)(a-b)(a3+ab2+ba2+b3-bc2-b3-cb2-c3)

= (b-c)(a-b)(a3+ab2+ba2-bc2-c3-cb2)
= (b-c)(a-b)(a3-c3)+b2(a-c)+b(a2-c2)
= (b-c)(a-b)(a-c)(a2+ac+c2)+b2(a-c)+b(a-c)(a+c)
= (b-c)(a-b)(a-c)(a2+ac+c2+b2+ab+ac)

= (a-b)(b-c)(c-a)(a2+b2+c2+ab+bc+ca)