Factor the following expression by grouping. \( r^{3}-2 r^{2}+4 r-8 \)

To factor the polynomial \( r^{3} - 2r^{2} + 4r - 8 \) by grouping, follow these steps: 1. **Group the terms:** \[ (r^{3} - 2r^{2}) + (4r - 8) \] 2. **Factor out the greatest common factor from each group:** \[ r^{2}(r - 2) + 4(r - 2) \] 3. **Factor out the common binomial factor \((r - 2)\):** \[ (r^{2} + 4)(r - 2) \] So, the factored form of the expression is: \[ (r^{2} + 4)(r - 2) \] *Note:* The quadratic \( r^{2} + 4 \) cannot be factored further using real numbers. If complex numbers are allowed, it can be factored as \( (r + 2i)(r - 2i) \), where \( i \) is the imaginary unit. **Final Answer:** \[ (r^{2} + 4)(\, r\ -\ 2\,) \]