Select the correct factors for the function: \( g(x)=x^{3}-4 x^{2}+3 x-12 \) \( x^{2}(x-4)(x+3) \) \( (x-4)\left(x^{2}+3\right) \) \( (x-3)(x+2)^{2} \) \( (x-3)\left(x^{2}+4\right) \)

To factor the function \( g(x) = x^{3} - 4x^{2} + 3x - 12 \), we can apply the Rational Root Theorem and synthetic division to test potential rational roots. Testing \( x = 3 \) yields a factor, because \( g(3) = 0 \). Thus, we can factor \( g(x) \) as follows: First, we divide \( g(x) \) by \( x - 3 \), leading us to \( g(x) = (x - 3)(x^2 - x - 4) \). Now, \( x^2 - x - 4 \) does not easily factor into rational terms, but we can find it has complex roots. So, the valid factorization of \( g(x) \) over the reals can be kept as is. Therefore, the correct option here is: \( (x-3)\left(x^{2}+4\right) \). Snugly bundled, the second one \( (x-4)\left(x^2+3\right) \) may be tempting, yet it doesn’t satisfy the roots derived from synthetic division. Remember: Always verify possible roots first!