Answer
$10+11i$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the FOIL Method, the given expression, $
(3+\sqrt{-4})+(4+\sqrt{-1})
,$ is equivalent to
\begin{align*}
&
3(4)+3(\sqrt{-1})+\sqrt{-4}(4)+\sqrt{-4}(\sqrt{-1})
\\&=
12+3(\sqrt{-1})+4\sqrt{-4}+\sqrt{-4}(\sqrt{-1})
.\end{align*}
Using $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$, the expression above is equivalent to
\begin{align*}
&
12+3(\sqrt{-1})+4\sqrt{-1}\cdot\sqrt{4}+\sqrt{-1}\cdot\sqrt{4}(\sqrt{-1})
\\&=
12+3(\sqrt{-1})+4\sqrt{-1}\cdot2+\sqrt{-1}\cdot2(\sqrt{-1})
\\&=
12+3(\sqrt{-1})+8\sqrt{-1}+2\sqrt{-1}(\sqrt{-1})
.\end{align*}
Since $i=\sqrt{-1},$ the expression above is equivalent to
\begin{align*}
&
12+3i+8i+2i(i)
\\&=
12+3i+8i+2i^2
\\&=
12+3i+8i+2(-1)
&\text{ (use $i^2=-1$)}
\\&=
12+3i+8i-2
.\end{align*}
Combining the real parts and the imaginary parts, the expression above is equivalent to
\begin{align*}
&
(12-2)+(3i+8i)
\\&=
10+11i
.\end{align*}
