#### Answer

$-9+40i$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
(4+5i)^2
,$ use the special product on squaring binomials. Then use $i^2=-1$ and combine like terms.
$\bf{\text{Solution Details:}}$
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(4)^2+2(4)(5i)+(5i)^2
\\\\=
16+40i+25i^2
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
16+40i+25(-1)
\\\\=
16+40i-25
.\end{array}
Combining like terms results to
\begin{array}{l}\require{cancel}
(16-25)+40i
\\\\=
-9+40i
.\end{array}