Which expression is the result of the perimeter of rectangle b minus the perimeter of rectangle a?

the following table shows the length and width of two rectangles:

length width
rectangle a x + 8 x − 1
rectangle b 4x + 5 3x − 2

which expression is the result of the perimeter of rectangle b minus the perimeter of rectangle a?
10x − 12
18x + 32
10x − 8
18x + 6

10x - 8

Step-by-step explanation:

The perimeter of these can be found by adding length and width and then multiplying by two. 

Rectangle A: (with all of the variables already doubled)

2x + 16 + 2x - 2 

2x + 2x + 16 - 2

4x + 14

So rectangle A's perimeter is 4x + 14.

Rectangle B: (Still with all the variables already doubled)

8x + 10 + 6x - 4

8x + 6x + 10 - 4

14x + 6

So rectangle B's perimeter is 14x + 6

And now to subtract the two.

(7x + 3) - (4x + 14)

14x + 6 - 4x - 14

14x - 4x + 6 - 14

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4