Factor $(ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3$ as much as possible.

Answer 1

We can simplify the given expression as follows:

$(ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3$

Expand the left side using the formula $(x+y+z)^3 = x^3 + y^3 + z^3 + 3xy(x+y) + 3xz(x+z) + 3yz(y+z)$:

$(ab)^3 + (ac)^3 + (bc)^3 + 3(ab)(ac)(ab + ac) + 3(ab)(bc)(ab + bc) + 3(ac)(bc)(ac + bc) - a^3 b^3 - a^3 c^3 - b^3 c^3$

$= a^3 b^3 + a^3 c^3 + b^3 c^3 + 3a^2 b^2 c + 3a^2 b c^2 + 3a b^2 c^2 - a^3 b^3 - a^3 c^3 - b^3 c^3$

$= 3a^2 b^2 c + 3a^2 b c^2 + 3a b^2 c^2$

Therefore, the factored form of the given expression is $\boxed{3abc(a^2+b^2+c^2)}$.