1. **Problem:** Find the roots of the polynomial $f(x) = x^4 - 10x^3 + 20x^2 + 19x - 12$. 2. **Step 1: Use synthetic division to factor the polynomial.** Given synthetic division steps show $f(x)$ divided by $(x+1)$ yields quotient $x^3 - 11x^2 + 31x - 12$. 3. **Step 2: Factor the quotient further.** The quotient $x^3 - 11x^2 + 31x - 12$ can be factored as $(x-1)(x^2 - 7x - 2)$. 4. **Step 3: Solve the quadratic $x^2 - 7x - 2 = 0$.** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-7$, $c=-2$. 5. **Step 4: Calculate the discriminant and roots.** $$\sqrt{(-7)^2 - 4 \times 1 \times (-2)} = \sqrt{49 + 8} = \sqrt{57}$$ 6. **Step 5: Write the roots explicitly.** $$x = \frac{7 \pm \sqrt{57}}{2}$$ 7. **Step 6: Collect all roots of $f(x)$.** From factorization: - Root from $(x+1)=0$ is $x = -1$. - Root from $(x-1)=0$ is $x = 1$. - Roots from quadratic are $x = \frac{7 \pm \sqrt{57}}{2}$. **Final answer:** $$x = -1, \quad x = 1, \quad x = \frac{7 + \sqrt{57}}{2}, \quad x = \frac{7 - \sqrt{57}}{2}$$