Answer
Length $9$ feet.
Width $6$ feet.
Work Step by Step
Let the width of the rectangular sign be $w=x$.
Therefore the length of the rectangular sign is $l=x+3$.
Area of the rectangular sign is $A=54$ square feet.
Formula for the area of the rectangular sign is
$A=l\times w$
Plug all values.
$54=(x)(x+3)$
Use the distributive property on the right hand side.
$54=x^2+3x$
Subtract $54$ from both sides.
$54-54=x^2+3x-54$
Simplify.
$0=x^2+3x-54$
Rewrite the middle term $3x$ as $9x-6x$.
$0=x^2+9x-6x-54$
Group terms.
$0=(x^2+9x)+(-6x-54)$
Factor out from each term.
$0=x(x+9)-6(x+9)$
Factor out $(x+9)$.
$0=(x+9)(x-6)$
Set both factors equal to zero.
$x+9=0$ or $x-6=0$
Isolate $x$.
$x=-9$ or $x=6$
Take the positive value $x=6$ as $x$ is a dimension.
Therefore width $w=6$ feet.
Length $l=x+3$
$l=6+3$
$l=9$ feet.