can not figure this one out for the life of me

Question

1. Nicholas paints a masterpiece on a canvas such that the width is 6 inches longer than its height. What would be the equation for the area of his canvas (written in simplified form)?
2. Nicholas added a 4-inch silver frame around all sides of his painting. What would be the new equation for the area of his canvas with the frame (written in simplified form)?
3. Nicholas ended up not liking the silver frame and replaced it with a gold frame. The area of the painting with the new frame can be represented with the polynomial h²+ 12h + 27, where h represents the height of the painting. Determine the width of the new gold frame.

Answer 1

W=6+H

A=WH = (6+H)H

A =6H +H^2

new A = (H+6+8)(H+8) = (H+14)(H+8)

= H^2 +22H +112

area of new frame + original = H^2 +12H+ 27

area of just the new frame = new total area - original =H^2 +12H +27 - 6H-H^2

= 6H +27

Answer 2

1) Since the width is 6 inches longer than the height, we can express the width as w = 6 + h. The equation for the area of a rectangle is w*h so the area of this canvas will be (h+6)(h) = h^2 + 6h.

2) when Nicholas adds in the silver frame, there are additional rectangles, namely 2 with area 4(6+h) and 2 with area 4h, and then 4 squares of area 16. Thus we can simply add these areas together.

Area of canvas + Area of frame = (h+6)(h) + (2)(4)(6+h) + 2(4)(h) + 4(16) = h^2 + 22h + 112.

3) let's express the width of the frame as a variable. When we do this, we will still have:

Area of canvas + Area of frame = (h+6)(h) + 2r(6+h) + 2rh + 4r^2 = h^2+6h+12r+4hr+4r^2.

This is what we have, and we want to find a value of r such that

h^2 + 6h + 4hr + 12r + 4r^2 = h^2 + h(6 + 4r) + 12r + 4r^2 turns into h²+ 12h + 27.

We can see that this will be true then: 6+4r = 12, and 12r + 4r^2 = 27.

Let's solve for r with the linear equation We attain that r = 1.5.

Lets solve for r in the quadratic 12r + 4r^2 = 27. When we apply the quadratic formula we see r = -4.5, 1.5.

r cannot be negative so we toss out -4.5.

Furthermore, notice that r = 1.5 is repeated in both! This is the answer.

So to recap,

1) Express the width in terms of the height, then multiply them together since the width of a rectangle is width*height.

2) add in the additional area gained from the frame. This will be 4 rectangles and 4 squares (draw it out and you will see)

3) Express the length of the frame as another variable. Write out the general equation, match terms, and solve the resulting system of equations. In general, if everything is done correctly you should get two matching solutions.

Hope this helps!