Which of the following is a factor of (x + y)^3 - (x^3 + y^3)?

For any two terms a and b, the binomial expansion gives (a + b)3 = a3 + 3a2b + 3ab2 + b3. Subtracting the sum of cubes (a3 + b3) from this expansion cancels the two cubic terms and leaves only the two middle cross-terms, giving the general identity (a + b)3 - (a3 + b3) = 3ab(a + b).

1. Expand (x + y)3 using the identity above: (x + y)3 = x3 + 3x2y + 3xy2 + y3.

2. Subtract (x3 + y3) from this expansion: (x3 + 3x2y + 3xy2 + y3) - (x3 + y3) = 3x2y + 3xy2.

3. Factor the common term out of 3x2y + 3xy2: both terms share the factor 3xy, so 3x2y + 3xy2 = 3xy(x + y).

4. Since the simplified expression equals 3xy(x + y), dividing it by 3xy leaves (x + y) with no remainder, so 3xy is a factor of the expression.

As an independent check, substitute x = 1 and y = 1: (1 + 1)3 - (13 + 13) = 8 - 2 = 6, and 3xy(x + y) at these values gives 3(1)(1)(2) = 6 — the two results match, confirming the factorization.