Question

Do any of you know how to do this

Answer 1

Answer:

Step-by-step explanation:

I certainly do know how to do this.  I would normally steer you towards long division of polynomials, but doing that here in this forum is quite literally impossible.  But synthetic division works!  So we will use that method of division of polynomials instead.  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.  Right?  Same thing here.  We will divide the quadratic by the width x + 3 to find the length.  

In the world of quadratics, if x + 3 is a 0 of the quadratic, then x + 3 = 0 and

x = -3.  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|    1    8    15

   __________

The first thing to do is bring down the first number and multiply it by the number in the box.  We will bring down the 1 and multiply it by -3 and put the product up under the 8:

-3|    1    8    15

            -3

     _________

      1

Now we will add:

-3|     1     8     15

              -3

     -------------------

       1     5

Multiply -3 by 5 and put that product up under the 15:

-3|     1     8     15

              -3    -15

    ---------------------

       1      5  

Now we will add:

-3|     1      8     15

               -3   -15

    ---------------------

        1       5      0

Because we got a 0 remainder, we know that x + 3 divides into the quadratic evenly.  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.  We started with a second degree, so the depressed polynomial is a first degree (linear) polynomial.  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!

Answer 2

Answer:

$(x + 5)$

Step-by-step explanation:

${x}^{2} + 8x + 15 \\ {x}^{2} + 3x + 5x + 15 \\ x(x + 3)5(x + 3) \\ (x + 3)(x + 5) \\ length \: = \frac{(x + 3)(x + 5)}{(x + 3)} \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = (x + 5)$