1. **State the problem:** We are given a polynomial $$P(x) = x^3 - 4x^2 - x + 4$$ and told that $$(x-1)$$ is a factor. We need to express $$P(x)$$ as a product of factors. 2. **Recall the factor theorem:** If $$(x - a)$$ is a factor of $$P(x)$$, then $$P(a) = 0$$. 3. **Verify that $$(x-1)$$ is a factor:** Substitute $$x=1$$ into $$P(x)$$: $$P(1) = 1^3 - 4(1)^2 - 1 + 4 = 1 - 4 - 1 + 4 = 0$$ Since $$P(1) = 0$$, $$(x-1)$$ is indeed a factor. 4. **Divide $$P(x)$$ by $$(x-1)$$ to find the other factor:** Use polynomial division or synthetic division. Synthetic division setup: Coefficients: 1 (for $$x^3$$), -4 (for $$x^2$$), -1 (for $$x$$), 4 (constant) Divide by root $$x=1$$: Carry down 1. Multiply 1 * 1 = 1, add to -4 = -3. Multiply 1 * -3 = -3, add to -1 = -4. Multiply 1 * -4 = -4, add to 4 = 0 (remainder). So the quotient is $$x^2 - 3x - 4$$. 5. **Factor the quadratic $$x^2 - 3x - 4$$:** Find two numbers that multiply to $$-4$$ and add to $$-3$$: these are $$-4$$ and $$1$$. So, $$x^2 - 3x - 4 = (x - 4)(x + 1)$$. 6. **Write the full factorization:** $$P(x) = (x - 1)(x - 4)(x + 1)$$. 7. **Check the options:** The correct factorization is $$(x - 1)(x - 4)(x + 1)$$. **Final answer:** $$(x - 1)(x - 4)(x + 1)$$