Sharon is paving a rectangular concrete driveway on the side of her house. The area of the driveway is 5x2 + 43x - 18, and the length of the driveway is x + 9. Additionally, Sharon plans to install a carport over a small portion of the driveway. The volume that the carport can enclose is 48x3 + 68x2 - 8x - 3, and the area of driveway beneath the carport is 8x2 + 10x - 3. Determine the width of the entire driveway and height of the carport in terms of x. Replace the values of m and b to complete the expression that represents the width of the entire driveway on the first line, and then replace the values of m and b to complete the expression that represents the height of the carport on the second line.

The width of the driveway is 5x-2 and the height of the carport is 6x+1.

The area of a rectangle is given by length * width.  Since the area is 5x²+43x-18 and the length is x+9, to find the width, we divide:

The first term of the quotient will be the number of times x goes into 5x².  It goes 5x times; multiply this by the divisor, and we have 5x²+45x to write under the first two terms of the dividend.

Subtracting these, we have -2x left over; bring down the -18.  The second term of the quotient will be the number of times x+9 goes into -2x-18; it goes -2 times.  Multiply this by the divisor, and we have -2x-18 with no remainder.  This gives the quotient 5x-2.

The volume of the carport is given by the area of the base * the height; since the volume is 48x³+68x²-8x-3 and the area of the base is 8x²+10x-3, we divide to get the height.

The first term of the quotient will be the number of times 8x² goes into 48x³.  It goes 6x times; multiply this by the divisor and we have 48x³+60x² to write under the first two terms of the dividend.

Subtract these and we have 8x²+10x left over; bring down the -3.  The second term of the quotient will be the number of times 8x² goes into 8x²; it goes 1 time.  Multiply this by the divisor and we have 8x²+10x-3 with no remainder.  This makes the height 6x-3.