Answer
$-50+40i$
Work Step by Step
Using $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ the given expression, $
(4+\sqrt{-25})(\sqrt{-100})
,$ is equivalent to
\begin{align*}
&
(4+\sqrt{-1}\cdot\sqrt{25})(\sqrt{-1}\cdot\sqrt{100})
\\&=
(4+\sqrt{-1}\cdot5)(\sqrt{-1}\cdot10)
\\&=
(4+i\cdot5)(i\cdot10)
&\text{ (use $i=\sqrt{-1}$)}
\\&=
(4+5i)(10i)
\\&=
4(10i)+5i(10i)
&\text{ (use Distributive Property)}
\\&=
40i+50i^2
\\&=
40i+50(-1)
&\text{ ($i^2=-1$)}
\\&=
40i-50
\\&=
-50+40i
.\end{align*}
Hence, the given expression simplifies to $
-50+40i
$.
