Answer
$-18+24i$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given expression, $
3i(-3-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}
3i[(-3)^2+2(-3)(-i)+(-i)^2]
\\\\=
3i[9+6i+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}
3i(9)+3i(6i)+3i(i^2)
\\\\=
27i+18i^2+3i(i^2)
.\end{array}
Since $i^2=-1,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
27i+18(-1)+3i(-1)
\\\\=
27i-18-3i
.\end{array}
Combining like terms results to
\begin{array}{l}\require{cancel}
-18+(27i-3i)
\\\\=
-18+24i
.\end{array}
