Answer
$8\times10 \text{ inches}$
Work Step by Step
Let $x$ be the width and $x+2$ be the length. Since if each dimension is increased by $4$ inches, the area is increased by $88,$ then
\begin{array}{l}\require{cancel}
A=lw
\\\\
x(x+2)+88=(x+4)(x+2+4)
\\\\
x^2+2x+88=(x+4)(x+6)
\\\\
x^2+2x+88=x(x)+x(6)+4(x)+4(6)
\\\\
x^2+2x+88=x^2+6x+4x+24
\\\\
x^2+2x+88=x^2+10x+24
\\\\
x^2-x^2+2x-10x=24-88
\\\\
-8x=-64
\\\\
x=\dfrac{-64}{-8}
\\\\
x=8
.\end{array}
Hence, the dimensions are $
8\times10 \text{ inches}
.$