# How do you simplify (4+2i)(-5+2i)?

Jun 23, 2016

$\left(4 + 2 i\right) \left(- 5 + 2 i\right) = - 24 - 2 i$

#### Explanation:

We simplify the multiplication of complex numbers by doing the multiplication like a binomial - i.e. we distribute the multiplication over the summed terms:

$\left(4 + 2 i\right) \left(- 5 + 2 i\right) = 4 \left(- 5\right) + 4 \left(2 i\right) + 2 i \left(- 5\right) + 2 i \left(2 i\right)$
$\left(4 + 2 i\right) \left(- 5 + 2 i\right) = - 20 + 8 i - 10 i + 4 {i}^{2}$

now we collect like terms and use ${i}^{2} = - 1$

$\left(4 + 2 i\right) \left(- 5 + 2 i\right) = - 20 - 2 i + 4 \left(- 1\right)$
$\left(4 + 2 i\right) \left(- 5 + 2 i\right) = - 20 - 2 i - 4$

Finally,

$\left(4 + 2 i\right) \left(- 5 + 2 i\right) = - 24 - 2 i$