Show that $a^3+a^2b+ab^2+b^3$ is not a prime, if a and b are positive integers

Question

I approached it by attempting to factor it and then show that one factor can't be one and the other can't be prime. This gets nowhere, as you can't factor this expression. Is there another way to do it?

Answer 1

Hint: $a^3+a^2b+ab^2+b^3=(a^2 + b^2)(a + b)$.

Answer 2

$a^3+b^3+a^2b+ab^2\\=(a+b)(a^2-ab+b^2)+a^2b+ab^2\\=(a+b)(a^2-ab+b^2)+ab(a+b)\\=(a+b)(a^2-ab+b^2+ab)\\=(a+b)(a^2+b^2)$

Answer 3

Hint: $(a-b)(a^3+a^2b+ab^2+b^3) = a^4-b^4=(a^2-b^2)(a^2+b^2)=(a-b)(a+b)(a^2+b^2)$