Answer
$-1+5i$
Work Step by Step
Since $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b},$ the given expression, $
(2+\sqrt{-1})+(-3+\sqrt{-16})
,$ is equivalent to
\begin{align*}
&
(2+\sqrt{-1})+(-3+\sqrt{-1}\cdot\sqrt{16})
\\&=
(2+\sqrt{-1})+(-3+\sqrt{-1}\cdot4)
\\&=
(2+\sqrt{-1})+(-3+4\sqrt{-1})
.\end{align*}
Since $i=\sqrt{-1},$ the expression above is equivalent to
\begin{align*}
(2+i)+(-3+4i)
.\end{align*}
Removing the grouping symbols, the expression above is equivalent to
\begin{align*}
2+i-3+4i
.\end{align*}
Combining the real parts and the imaginary parts, the expression above is equivalent to
\begin{align*}
&
(2-3)+(i+4i)
\\&=
-1+5i
.\end{align*}
Hence, the simplified form of the given expression is $
-1+5i
$.