1. **Problem statement:** Divide the polynomial $f(x) = x^3 + 4x^2 + x - 6$ by the linear factor $(x - 1)$. Find the quotient polynomial. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by this result, subtract, and repeat with the remainder. 3. **Step-by-step division:** Divide the leading term $x^3$ by $x$ to get $x^2$. Multiply $(x - 1)$ by $x^2$ to get $x^3 - x^2$. Subtract: $$\left(x^3 + 4x^2 + x - 6\right) - \left(x^3 - x^2\right) = \cancel{x^3} + 4x^2 + x - 6 - \cancel{x^3} + x^2 = 5x^2 + x - 6$$ Divide $5x^2$ by $x$ to get $5x$. Multiply $(x - 1)$ by $5x$ to get $5x^2 - 5x$. Subtract: $$\left(5x^2 + x - 6\right) - \left(5x^2 - 5x\right) = \cancel{5x^2} + x - 6 - \cancel{5x^2} + 5x = 6x - 6$$ Divide $6x$ by $x$ to get $6$. Multiply $(x - 1)$ by $6$ to get $6x - 6$. Subtract: $$\left(6x - 6\right) - \left(6x - 6\right) = \cancel{6x} - 6 - \cancel{6x} + 6 = 0$$ 4. **Result:** The quotient is $x^2 + 5x + 6$ with remainder $0$. 5. **Interpretation:** Since the remainder is zero, $(x - 1)$ is a factor of $f(x)$, and the division is exact. **Final answer:** $$f(x) \div (x - 1) = x^2 + 5x + 6$$