Factor the following by grouping. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( x^{3}-4 x^{2}+x-4 x^{2}+x-4=\square \) B. The polynomial is prime.

To factor the polynomial \( x^{3} - 4x^{2} + x - 4 \) by grouping, we will follow these steps: 1. **Group the terms**: We can group the first two terms and the last two terms. \[ (x^{3} - 4x^{2}) + (x - 4) \] 2. **Factor out the common factors in each group**: - From the first group \( x^{3} - 4x^{2} \), we can factor out \( x^{2} \): \[ x^{2}(x - 4) \] - From the second group \( x - 4 \), we can factor out \( 1 \): \[ 1(x - 4) \] 3. **Combine the factored groups**: Now we can write the expression as: \[ x^{2}(x - 4) + 1(x - 4) \] 4. **Factor out the common binomial factor**: Since both terms contain the factor \( (x - 4) \), we can factor that out: \[ (x - 4)(x^{2} + 1) \] Now, we have factored the polynomial. Therefore, the correct choice is: A. \( x^{3} - 4x^{2} + x - 4 = (x - 4)(x^{2} + 1) \)