Question

How do I️ solve this

Answer 1

While you didn't post the instructions for this problem, I can safely assume that you're supposed to factor 8x^2 - 8x - 30.

First, note that all of these terms can be divided by 2:

8x^2 - 8x - 30 = 2(4x^2 - 4x - 15)

Inside the parentheses is a quadratic equation which can't be reduced further.  There are a good number of methods that you could use to solve this quadratic:

graphing, completing the square, factoring, quadratic formula, and so on.

I note that the first and last coefficients are 4 and 15.

Thus, possible roots are:

numerator           1, 3, 5, 15, -1, -3, -5, -15
------------------ = -----------------------------------
denominator        1, 2, 4, -1, -2, -4

For example, one root might be -5/2 (-5 from the numerator and 2 from the denominator).

Let's check whether -5/2 is actually a soluion, using synthetic div:

            ___________________
  -5/2   /   4        -4      -15
                        -10      7/2
                  __________________
               4       -14      Remainder is NOT zero, so -5/2 is not a root.

Lots of possibilities here.

Thus, I'm going to resort to the good old Quadratic Formula to determine the 
factors of 4x^2 -4x - 15:

a=4, b= -4 and c = -15

Then the the roots are:

         4 plus or minus sqrt(16-4(4)(-15)
x = ------------------------------------------------
                               8
        4 plus or minus sqrt(256)        4 plus or minus 16
   = ------------------------------------- = -------------------------------
                                8                                 8

    = 20/8 and -12/8, or 5/2 and -3/2

Thus, in factored form the polynomial is 8(x-5/2)(x+3/2)