To solve the equation X(X^2 + 2X + 1) = 2(X + 1), we first need to expand both sides:

X(X^2 + 2X + 1) = 2(X + 1)
X^3 + 2X^2 + X = 2X + 2

Next, combine like terms on both sides of the equation:

X^3 + 2X^2 + X - 2X - 2 = 0
X^3 + 2X^2 - X - 2 = 0

Now, we need to factor the equation by grouping:

X^2(X + 2) - 1(X + 2) = 0
(X^2 - 1)(X + 2) = 0
(X - 1)(X + 1)(X + 2) = 0

Therefore, the solutions to the equation X(X^2 + 2X + 1) = 2(X + 1) are X = -2, X = -1, and X = 1.