To factor the expression x^4 - 2x^3 - 12x^2 + 12x + 36, we can first look for any common factors among the terms. In this case, all the terms have a common factor of 2, so we can factor it out:

2(x^4 - x^3 - 6x^2 + 6x + 18)

Next, we can try to factor the expression inside the parentheses. We can start by grouping the terms:

2((x^4 - x^3) + (-6x^2 + 6x) + 18)

Now, let's factor out the common factors from each group:

2(x^3(x - 1) - 6x(x - 1) + 18)

Notice that we have a common factor of (x - 1) in each group. We can factor it out:

2(x - 1)(x^3 - 6x + 18)

Therefore, the factored form of the expression x^4 - 2x^3 - 12x^2 + 12x + 36 is 2(x - 1)(x^3 - 6x + 18).