We have to verify if (a+b)^5 - a^5 - b^5 = 5ab(a+b)(a^2+ab+b^2)

We can expand (a + b)^5 = a^5 + 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5

The left hand side

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

=> a^5 + 5a^4b + 10 a^3b^2 + 10...

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We have to verify if (a+b)^5 - a^5 - b^5 = 5ab(a+b)(a^2+ab+b^2)

We can expand (a + b)^5 = a^5 + 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5

The left hand side

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

=> a^5 + 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4 + b^5 - a^5 - b^5

=> 5a^4b + 10 a^3b^2 + 10 a^2b^3 + 5ab^4

=> 5ab(a^3 + 2a^2b + 2ab^2 + b^3)

=> 5ab(a + b)( a^2 + ab + b^2)

which is the right hand side

**We prove that (a+b)^5 - a^5 - b^5 = 5ab(a+b)(a^2+ab+b^2).**