To factorize the expression x^3 + 9x^2 + 23x + 15 using the factor theorem, we need to find the possible factors of the constant term (15) and the leading coefficient (1) and test them using synthetic division.

Step 1: Possible Factors
The possible factors of 15 are ±1, ±3, ±5, and ±15.
The possible factors of 1 are ±1.

Step 2: Synthetic Division
We will test each possible factor using synthetic division to find the remainder.

Testing x = 1:
1 | 1 9 23 15
| 1 10 33
| -----------------
1 10 33 48

The remainder is 48.

Testing x = -1:
-1 | 1 9 23 15
| -1 -8 -15
| -----------------
1 8 15 0

The remainder is 0, which means (x + 1) is a factor.

Step 3: Factorization
Since (x + 1) is a factor, we can divide the original expression by (x + 1) using synthetic division to obtain the remaining factor.

-1 | 1 9 23 15
| -1 -8 -15
| -----------------
1 8 15 0

The result is x^2 + 8x + 15.

Now, we can factorize the remaining quadratic expression x^2 + 8x + 15 by finding its factors.

Step 4: Factorizing the Quadratic Expression
The factors of the quadratic expression can be found by factoring the quadratic equation x^2 + 8x + 15 = 0.

(x + 5)(x + 3) = 0

Therefore, the factorization of the expression x^3 + 9x^2 + 23x + 15 using the factor theorem is:
(x + 1)(x + 5)(x + 3).

Final Factorization:
The expression x^3 + 9x^2 + 23x + 15 can be factorized as:
(x + 1)(x + 5)(x + 3).

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