A classroom will be assembled so that the desks fit within an area that is 8 m by 12 m. The desks will be surrounded by a border that will be the same width on all sides of the rectangular room, to allow for walking space. The area of the border will equal 20% of the area of the space that the desks
occupy. What is the width of the border, to the nearest hundredth of a meter?

0.46 m

Step-by-step explanation:

area of desks: 8 m by 12 m = 8 m * 12 m = 96 m^2

20% of this area is 20% * 96 m^2 = 19.2 m^2

The area of the border is 19.2 m^2.

Let the path around the desks have width x.

The area of desks plus path is a rectangle 2x + 8 by 2x + 12.

area of desks plus path = (2x + 8)(2x + 12)

= 4x^2 + 24x + 16x + 96 = 4x^2 + 40x + 96

The area of the border is the area of the rectangle that includes the border minus the rectangle that has just the desks.

area of border = (4x^2 + 40x + 96) - (96) =

= 4x^2 + 40x

Above, we have the area of border = 19.2, so we get this equation:

4x^2 + 40x = 19.2

4x^2 + 40x - 19.2 = 0

x^2 + 10x - 4.8 = 0

We now use the quadratic formula to solve the equation for x.

 or  

We discard the negative solution.

the border is 0.46 m wide.

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."