You have a rectangle with the following expressions for the area and the width:
Area: A = 14c² - c - 4
width: w = 2c + 1
Take into accoint that the area of a rectangle is the product between its width and length.
A = w·l
if you divide the area over the width you obtain the length:
l = A/w
then, divide the expressions for A and w, in standard division form, as follow:
14c² - c - 4 | 2c + 1
-14c²-7c 7c - 4
-8c - 4
8c +4
0
then, the result is 7c - 4, which is the expression for the length of the rectangle:
length: l = 7c - 4
Now, consider that the perimeter of the rectangle is given by:
P = 2w + 2l
replace the expressions for w and l:
P = (2c + 1) + (7c - 4) open parenthesis
P = 2c + 7c +1 - 4
P = 9c - 3
Hence, the permeter