Answer
$x=\left\{ 3 \right\}$
Work Step by Step
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the Distributive Property, the given expression, $
(x-7)(x+1)=-16
,$ is equivalent to
\begin{array}{l}\require{cancel}
x(x)+x(1)-7(x)-7(1)=-16
\\\\
x^2+x-7x-7=-16
\\\\
x^2+x-7x-7+16=0
\\\\
x^2-6x+9=0
.\end{array}
Factoring the above equation, $
x^2-6x+9=0
,$ results to
\begin{array}{l}\require{cancel}
x^2-6x+9=0
\\\\
(x-3)^2=0
\\\\
(x-3)(x-3)=0
.\end{array}
Equating each factor to zero (Zero Product Principle), then the solutions to the equation, $
(x-3)(x-3)=0
,$ are
\begin{array}{l}\require{cancel}
x-3=0
\\\\
x=3
,\\\\\text{OR}\\\\
x-3=0
\\\\
x=3
.\end{array}
Hence, $
x=\left\{ 3 \right\}
.$
