\( 1 \leftarrow \) Fing Polynomilals \[ x^{3}-2 x^{2}-4 x+8 \] Select the correct choice below and, if necessary, fill in the answer box within your choice. A. \( x^{3}-2 x^{2}-4 x+8=\square \) B. The polynomial is prime.

We start with the polynomial \[ x^{3} - 2x^{2} - 4x + 8. \] **Step 1: Grouping Terms** Group the polynomial into two pairs: \[ \begin{aligned} x^{3} - 2x^{2} - 4x + 8 &= (x^{3} - 2x^{2}) + (-4x + 8). \end{aligned} \] **Step 2: Factor Each Group** Factor out the greatest common factor (GCF) from each group: \[ \begin{aligned} x^{3} - 2x^{2} &= x^{2}(x - 2), \\ -4x + 8 &= -4(x - 2). \end{aligned} \] Now the expression becomes: \[ x^{2}(x-2) - 4(x-2). \] **Step 3: Factor Out the Common Binomial** Notice that \((x-2)\) is a common factor: \[ x^{2}(x-2) - 4(x-2) = (x-2)(x^{2} - 4). \] **Step 4: Factor the Quadratic Expression** Recognize that \(x^{2} - 4\) is a difference of squares: \[ x^{2} - 4 = (x-2)(x+2). \] **Step 5: Write the Final Factored Form** Substitute back into the expression: \[ (x-2)(x^{2} - 4) = (x-2)(x-2)(x+2) = (x-2)^{2}(x+2). \] Thus, the factorization is \[ x^{3} - 2x^{2} - 4x + 8 = (x-2)^{2}(x+2). \] Since the polynomial factors completely, the correct choice is: A. \( x^{3} - 2x^{2} - 4x + 8 = (x-2)^{2}(x+2) \).