To determine if $x = 2$ is a solution of the equation $x^3 + 4x^2 - 2x - 8 = 0$, we will use synthetic division. Then, we will factor the polynomial completely and find all real solutions.

Step 1: Synthetic Division

We'll divide the polynomial $x^3 + 4x^2 - 2x - 8$ by $x - 2$ using synthetic division.

1. Write down the coefficients of the polynomial $x^3 + 4x^2 - 2x - 8$:

   • Coefficients: $1, 4, -2, -8$

2. Set up the synthetic division with $x = 2$ as follows:

2 & 1 & 4 & -2 & -8 \\ & & 2 & 12 & 20 \\ \hline & 1 & 6 & 10 & 0 \\ \end{array}$$ ### Step 2: Explanation of the Synthetic Division Process 1. Bring down the first coefficient, $$1$$, and place it under the line. 2. Multiply $$2$$ (the value of $$x$$) by $$1$$, which gives $$2$$, and write it under the next coefficient. 3. Add $$4 + 2 = 6$$. 4. Multiply $$2 \times 6 = 12$$ and write it under the next coefficient. 5. Add $$-2 + 12 = 10$$. 6. Multiply $$2 \times 10 = 20$$, and write it under the last coefficient. 7. Add $$-8 + 20 = 0$$, which is the remainder. Since the remainder is zero, $$x = 2$$ is a solution, and the quotient from the division is $$x^2 + 6x + 10$$. ### Step 3: Factor the Polynomial The result from the synthetic division gives us: $$x^3 + 4x^2 - 2x - 8 = (x - 2)(x^2 + 6x + 10)$$ Next, we attempt to factor $$x^2 + 6x + 10$$. However, this quadratic does not factor nicely using real numbers, so we apply the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = 1$$, $$b = 6$$, and $$c = 10$$. $$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(10)}}{2(1)} = \frac{-6 \pm \sqrt{36 - 40}}{2} = \frac{-6 \pm \sqrt{-4}}{2} = \frac{-6 \pm 2i}{2}$$ $$x = -3 \pm i$$ So, the quadratic $$x^2 + 6x + 10$$ has complex roots $$x = -3 + i$$ and $$x = -3 - i$$. ### Step 4: List All Real Solutions The only real solution is $$x = 2$$, and the complete factorization of the polynomial is: $$(x - 2)(x^2 + 6x + 10)$$ ### Final Answer: The real solution is: $$x = 2$$ Do you want more details or have any questions? Here are some related questions for further understanding: 1. How do you interpret complex roots geometrically? 2. What is the significance of the remainder being zero in synthetic division? 3. How do you determine whether a quadratic has real or complex solutions? 4. Can synthetic division be used to divide by any polynomial, or just linear ones? 5. How would you verify the factorization of a cubic polynomial? **Tip:** When using the quadratic formula, always check the discriminant $$\Delta = b^2 - 4ac$$ to determine the nature of the roots (real or complex).