Answer
$2$ inches
Work Step by Step
The area of the rectangular board and the border is $A_1=(2x+12)(2x+16),$ while the area of the rectangular board is $A_2=12(16).$ Subtracting the two areas results to the area of the border. Since, the area of the border is given as $128$ square inches, then
\begin{array}{l}\require{cancel}
A=A_1-A_2
\\
128=(2x+12)(2x+16)-12(16)
\\
128=(2x+12)(2x+16)-192
\\
128+192=(2x+12)(2x+16)
\\
320=(2x+12)(2x+16)
.\end{array}
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
320=2x(2x)+2x(16)+12(2x)+12(16)
\\
320=4x^2+32x+24x+192
\\
0=4x^2+32x+24x+192-320
\\
4x^2+56x-128=0
\\
\dfrac{4x^2+56x-128}{4}=\dfrac{0}{4}
\\
x^2+14x-32=0
\\
(x+16)(x-2)=0
.\end{array}
Equating each factor to zero (Zero Product Property) then the solutions of the equation above are $
x=\{-16,2\}
.$ Since $x$ is a measurement, then only $x=2$ is the acceptable solution. Therefore, the width of the border, $x,$ is $2$ inches.
