# Question #84ae9

Jan 2, 2018

${x}^{8} - 4 {x}^{7} - 2 {x}^{6} + 20 {x}^{5} + {x}^{4} - 40 {x}^{3} - 8 {x}^{2} + 32 x + 16$

#### Explanation:

First, rearrange ${\left(2 + x - {x}^{2}\right)}^{4}$ into ${\left(- {x}^{2} + x + 2\right)}^{4}$. Then we have:

${\left(- {x}^{2} + x + 2\right)}^{4} = \left(- {x}^{2} + x + 2\right) \left(- {x}^{2} + x + 2\right) \left(- {x}^{2} + x + 2\right) \left(- {x}^{2} + x + 2\right)$

$= \left({x}^{4} - 2 {x}^{3} - 3 {x}^{2} + 4 x + 4\right) \left(- {x}^{2} + x + 2\right) \left(- {x}^{2} + x + 2\right)$

$= \left(- {x}^{6} + 3 {x}^{5} + 3 {x}^{4} - 11 {x}^{3} - 6 {x}^{2} + 12 x + 8\right) \left(- {x}^{2} + x + 2\right)$

$= {x}^{8} - 4 {x}^{7} - 2 {x}^{6} + 20 {x}^{5} + {x}^{4} - 40 {x}^{3} - 8 {x}^{2} + 32 x + 16$