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)
