Step-by-step explanation:
I certainly do know how to do this. Â I would normally steer you towards long division of polynomials, but doing that here in this forum is quite literally impossible. Â But synthetic division works! Â So we will use that method of division of polynomials instead. Â Ok?
First of all, if the area of a rectangle is, for example, 18 sq ft, and a width is 3, then by the area of a rectangle, A = LW, then 18 = 3L and by division, we know that L has to be 6. Â Right? Â Same thing here. Â We will divide the quadratic by the width x + 3 to find the length. Â
In the world of quadratics, if x + 3 is a 0 of the quadratic, then x + 3 = 0 and
x = -3. Â We will put -3 in a little box and then, in descending order, the coefficients of the quadratic which are 1, 8, and 15:
-3| Â Â 1 Â Â 8 Â Â 15
 Â
The first thing to do is bring down the first number and multiply it by the number in the box. Â We will bring down the 1 and multiply it by -3 and put the product up under the 8:
-3| Â Â 1 Â Â 8 Â Â 15
      -3
  Â
   1
Now we will add:
-3| Â Â 1 Â Â 8 Â Â 15
       -3
  Â
    1   5
Multiply -3 by 5 and put that product up under the 15:
-3| Â Â 1 Â Â 8 Â Â 15
       -3   -15
 Â
    1    5 Â
Now we will add:
-3| Â Â 1 Â Â Â 8 Â Â 15
        -3  -15
 Â
    1    5    0
Because we got a 0 remainder, we know that x + 3 divides into the quadratic evenly. Â The numbers there on the bottom, the 1 and the 5, give us the depressed polynomial, which is a degree less than the polynomial we started with. Â We started with a second degree, so the depressed polynomial is a first degree (linear) polynomial. Â Those numbers are the coefficients of the depressed polynomial, which is:
x + 5.
That means the length of the rectangle is x + 5.
If you multiply the width, x + 3, times the length, x + 5, which is the definition of area, you will get the second degree polynomial you started out with as the area.
I hope this helped and didn't confuse!