Hello Heidi, this is a great question.
To solve problems like these, you need to translate the words into equations.
"The room has an area of 176 square feet." means
Area = 176
Area of a rectangle is also length times width which gives:
l * w = 176
"and is 5 feet longer than it is wide" means
length = width + 5
l = w + 5
Now that we have both equations we can use substitution to get that the area is:
w * (w + 5) = 176
Even though you've tagged this problem as completing the square, I think it makes more sense to solve it using factoring techniques. Distributing the w gives:
w2 + 5w = 176
Rearranging to get the equation in standard form gives:
w2 + 5w - 176 = 0
Now, 176 does not factor extremely easily. There are a lot of different factoring techniques you can use. When I divide 176 by 2, I see that 2 and 88 are factors. So I have that 2*8*11= 176 or 16*11=176. This pair meets the requirements of the problem where the room is 11 feet wide and 16 feet long. Looking at it mathematically:
(w + 16)(w - 11) = 0
The positive root of this equation is w = 11. Returning to the equation that length = width + 5 gives that length = 11 + 5, which returns 16 feet.