Answer
$-16+30i$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
2i(-4-i)^2
,$ use the special product on squaring binomials and the Distributive Property. 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}
2i[(-4)^2+2(-4)(-i)+(-i)^2]
\\\\=
2i[16+8i+i^2]
.\end{array}
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2i(16)+2i(8i)+2i(i^2)
\\\\=
32i+16i^2+2i(i^2)
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
32i+16(-1)+2i(-1)
\\\\=
32i-16-2i
.\end{array}
Combining like terms results to
\begin{array}{l}\require{cancel}
-16+(32i-2i)
\\\\=
-16+30i
.\end{array}
