To factorize the given expression 6x^3 - 25x^2 + 32x - 12 using the factor theorem, we need to find the roots of the polynomial. The factor theorem states that if a polynomial has a factor (x - a), then when x = a, the polynomial will equal zero.

Step 1: Finding the potential roots

To find the potential roots, we can use the rational root theorem. According to the theorem, the possible rational roots of a polynomial are of the form p/q, where p is a factor of the constant term (in this case, -12) and q is a factor of the leading coefficient (in this case, 6).

The factors of -12 are ±1, ±2, ±3, ±4, ±6, and ±12.
The factors of 6 are ±1, ±2, ±3, and ±6.

From these lists, we can identify the potential rational roots as follows:

±1, ±2, ±3, ±4, ±6, and ±12 (divided by) ±1, ±2, ±3, and ±6

Step 2: Testing the potential roots

We can use synthetic division or substitute the potential roots into the polynomial to test them. If the result is zero, then the potential root is a root of the polynomial.

Let's start by testing x = 1:
f(1) = 6(1)^3 - 25(1)^2 + 32(1) - 12
= 6 - 25 + 32 - 12
= 1

Since f(1) ≠ 0, x = 1 is not a root of the polynomial.

Step 3: Applying the factor theorem

To factorize the polynomial using the factor theorem, we need to find a root that makes the polynomial equal to zero. Unfortunately, none of the potential rational roots we tested in Step 2 are roots of the polynomial.

Therefore, we can conclude that the polynomial 6x^3 - 25x^2 + 32x - 12 does not have any rational roots, and thus, it cannot be factorized using the factor theorem.

However, it is important to note that this does not mean the polynomial cannot be factorized further. There might exist irrational or complex roots that are not captured by the rational root theorem.

Hence, the factorization of 6x^3 - 25x^2 + 32x - 12 using the factor theorem is not possible with rational roots.