# How do you simplify 5(x - 4)(x + 1 + i)(x - 1 - i)?

May 22, 2018

$5 {x}^{3} - 20 {x}^{2} - 10 x i + 40 i$

#### Explanation:

$\text{expand the factors in pairs and multiply the expansions}$

$\Rightarrow 5 \left(x - 4\right) = 5 x - 20$

$\left(\left(x + 1\right) + i\right) \left(\left(x - 1\right) - i\right) \leftarrow \textcolor{b l u e}{\text{expand using FOIL}}$

$= \left(x + 1\right) \left(x - 1\right) - i \left(x + 1\right) + i \left(x - 1\right) - {i}^{2}$

$= {x}^{2} - 1 - x i - i + x i - i + 1 \to {i}^{2} = - 1$

$= {x}^{2} - 2 i$

$\Rightarrow \left(5 x - 20\right) \left({x}^{2} - 2 i\right)$

$= 5 {x}^{3} - 10 x i - 20 {x}^{2} + 40 i$