verify if the relationship is true? (a+b)^5-a^5-b^5=5ab(a+b)(a^2+ab+b^2)

We'll expand the
binomial:

(a+b)^5=a^5+5a^4*b+10a^3*b^2+10a^2*b^3+5b^4*a+b^5

We'll
subtract a^5 and b^5 from expansion and we'll
get:

5a^4*b+10a^3*b^2+10a^2*b^3+5b^4*a

We'll
combine the middle terms and the extremes and we'll factorize
them:

5ab(a^3 + b^3) +
10a^2*b^2(a+b)

But the sum of cubes
is:

a^3  +b^3 = (a+b)(a^2 - ab +
b^2)

5ab(a+b)(a^2 - ab + b^2) +
10a^2*b^2(a+b)

We'll factorize by
5ab(a+b)

5ab(a+b)(a^2 - ab + b^2 +
2ab)

We'll combine like terms inside
brackets:

5ab(a+b)(a^2 + ab +
b^2)=5ab(a+b)(a^2 + ab + b^2) q.e.d.