Let x be the length and y be the width of the rectangle.
It is given that width of a rectangular garden is 9 feet longer than 4 times its length so it follows:

It is also given that the garden's perimeter is 178 feet so it follows:

Add (i) and (ii) to get:

Substitute x=16 in (ii) to get:

Hence it follows that:
Length = 16 feet
Width =73 feet.