A rectangle has two parallel sides 4y+8 and 20x-16. The other parallel side are x^2 and yFind the perimeter if one side has an odd integer length.

Multiplication by an even number makes an odd number
even 
so, if y+8 and 20x-16 are even we need y^2 and x^2  to be
odd.

We'll put the lengths of parallel sides in the
relations:

4y+8 = 20x-16
(1)

and

y=x^2
(2)

We'll work over the relation
(1):

We'll subtract 20x and 8 both
sides:

4y - 20x = -8 - 16

4y -
20x = -24

From (2), we'll have
y=x^2.

 4x^2- 20x = -24

We'll
add 24 both sides:

4x^2- 20x + 24 =
0

We'll divide by 4:

x^2 - 5x
+ 6 = 0

We'll apply quadratic
formula:

x1 =
[5+sqrt(25-24)]/2

x1 =
(5+1)/2

x1 = 3

x2 =
(5-1)/2

x2 = 2

We'll re-write
the quadratic as a product of linear factors:

 x^2 - 5x + 6
= (x-2)(x-3)

If x1 = 3 => y = x1^2 => y1 =
9

If x2= 2=> y = x2^2 => y2 =
4

We'll choose the odd length for y: y =
9.

The perimeter is:

P =
(4y+8) +(20x-16)+y+x^2

P = (4*9 + 8) + (20*3 - 16) + 9 +
9

P = 44 + 44 + 2*9

P = 88 +
18

P = 106
units